# Non-autonomous soliton hierarchies

Maciej Błaszak† ,  Krzysztof Marciniak‡ and  Błażej Szablikowski† †Faculty of
Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614
Poznań, Poland  ‡Department of Science and Technology, Linköping University,
Campus Norrköping, 601 74 Norrköping, Sweden

###### Abstract.

A formalism of systematic construction of integrable non-autonomous
deformations of soliton hierarchies is presented. The theory is formulated as
an initial value problem (IVP) for an associated Frobenius integrability
condition on a Lie algebra. It is showed that this IVP has a formal solution
and within the framewrok of two particular subalgebras of the hereditary Lie
algebra the explicit forms of this formal solution are presented. Finally,
this formalism is applied to Korteveg-de Vries, dispersive water waves and
Ablowitz-Kaup-Newell-Segur soliton hierarchies.

##  1\. Introduction

In this article we present a systematic method of deforming of commuting
hierarchies of _autonomous_ evolutionary flows, i.e. systems of evolutionary
PDE’s of the form

$u_{t_{n}}=K_{n}[u],\qquad n=1,2,\ldots,$ (1.1)

(where $u=u(x)=\left(u_{1}(x),\ldots,u_{N}(x)\right)^{T}$ and where each
$K_{n}[u]$ is some vector field depending on $u$ and a finite number of its
$x$-derivatives, but not explicitely on times (i.e. evolution parameters
$t_{i}$) and such that

$\left[K_{m},K_{n}\right]=0,\qquad m,n=1,2,\ldots,$ (1.2)

to the _non-autonomous_ hierarchies of evolutionary flows

$u_{t_{n}}=\mathbb{K}_{n}\left([u],x,t_{1},\ldots,t_{n}\right)\qquad
n=1,2,\ldots,$ (1.3)

that satisfy the Frobenius integrability condition

$\frac{\partial\mathbb{K}_{n}}{\partial
t_{m}}-\frac{\partial\mathbb{K}_{m}}{\partial
t_{n}}+\left[\mathbb{K}_{m},\mathbb{K}_{n}\right]=0,\qquad m,n=1,2,\ldots.$
(1.4)

In (1.1) and (1.3) $K_{n}$ and $\mathbb{K}_{n}$ are vector fields, depending
on $u$ and a finite number of its $x$-derivatives on some infinite-dimensional
functional manifold and $K_{n}$ do not depend explicitly on times $t_{i}$.
Note that in (1.3) we assume _triangular_ dependence of vector fields
$\mathbb{K}_{n}$ on times $t_{i}$ for $1\leqslant i\leqslant n$, which in
result simplifies the Frobenius condition (1.4) to

$\frac{\partial\mathbb{K}_{n}}{\partial
t_{m}}+\left[\mathbb{K}_{m},\mathbb{K}_{n}\right]=0,\qquad m<n.$ (1.5)

The condition (1.2) (which is nothing else than the Frobenius integrability
condition for the autonomous system (1.1)) means that the system (1.1) has a
common, multi-time solution through each initial condition
$u(x,0,0,\ldots)=u_{0}(x)$. Likewise, the Frobenius condition (1.4) means that
the system (1.3) has a common, multi-time solution through each initial
condition $u(x,0,0,\ldots)=u_{0}(x)$. If these compatibility conditions are
not met it makes no sense to consider the systems in (1.1) or the systems in
(1.3) as hierarchies; they are simply not compatible.

In order to highlight the main algebraic ingredients of our construction we
first (Section 2) formulate our theory in a more general framework, as an
initial-value problem (IVP) for $\mathcal{A}$-valued functions
$\mathbb{K}_{n}$ satisfying (1.5), where $\mathcal{A}$ is a non-abelian Lie
algebra. In Theorem 3 and Corollary 4 we show that this IVP has a formal
solution. Next, we present solutions of the IVP for particular subalgebras of
the hereditary Lie algebra [1, 2, 3] (Section 3) and then we apply these
results to soliton hierarchies (Section 4). In Section 4 we also find the
zero-curvature representations for the non-autonomous hierarchies (1.3) from
zero-curvature representations of the corresponding autonomous hierarchies
(1.1). We illustrate our method on three examples: KdV hierarchy (Section 5),
dispersive water wave (DWW) hierarchy in the framework of [4, 5] (Section 5)
and Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [6] (Section 7).

We believe that the results presented in this article are important as the
majority of research in the theory of integrable PDE’s focuses on autonomous
systems of type (1.1). To our best knowledge, the non-autonomous deformations
(1.3) of soliton hierarchies have not been previously studied. The usual
approach to non-autonomous soliton equations is to modify a single chosen
soliton equation by assuming some time-dependence of one or more of its
coefficients, see for example [7, 8] or [9].

This article was inspired by our results presented in [10, 11, 12], where we
have considered polynomial in times deformations of autonomous Liouville
integrable finite dimensional systems, i.e. systems of the form

$\displaystyle\frac{dx}{dt_{i}}=\pi dh_{i},\qquad h_{i}=h_{i}(x),\qquad
i=1,\ldots,n,$ $\displaystyle\left\\{h_{i},h_{j}\right\\}_{\pi}=0,\qquad
i,j=1,\ldots,n,$

on some $2n$-dimensional manifold equipped with a Poisson bivector $\pi$, to
non-autonomous Frobenius integrable systems

$\displaystyle\frac{dx}{dt_{i}}=\pi dH_{i},\qquad
H_{i}=H_{i}(x,t_{1},\ldots,t_{n}),\qquad i=1,\ldots,n,$
$\displaystyle\left\\{H_{i},H_{j}\right\\}_{\pi}+\frac{\partial
H_{i}}{\partial t_{j}}-\frac{\partial H_{j}}{\partial t_{i}}=0,\qquad
i,j=1,\ldots,n,$

on the same manifold; here $x$ denotes points on this manifold. Another
inspiration for this work was the article [13] in which the author constructed
non-autonomous KdV hierarchies from Painlevé systems that were obtained as
deformations of Stäckel separable systems. This construction, however, was
dependent on the dimension $n$ of the underlying Stäckel systems; increasing
$n$ led to a completely different finite hierarchy. This drawback is not
present in the theory we develop in this article.

##  2\. Frobenius integrability condition in Lie algebras

In this section we present a _formal_ solution to the Frobenius integrability
condition (1.5) formulated as an initial-value problem (IVP) for finite or
infinite sets $\left\\{\mathbb{K}_{0},\mathbb{K}_{1},\ldots\right\\}$ of
elements $\mathbb{K}_{n}$ that belong to a non-abelian Lie algebra
$\mathcal{A}$ and such that each element $\mathbb{K}_{n}$ depends on (at most)
$n+1$ real evolutionary parameters (times) $t_{i}$, so that

$\mathbb{K}_{n}=\mathbb{K}_{n}(t_{0},\ldots,t_{n}).$ (2.1)

Thus, all elements $\mathbb{K}_{n}$ will be some $\mathcal{A}$-valued
functions of a finite number of real parameters $t_{i}$. The word _formal_
means in this context that we do not consider any convergence issues that may
arise in the formulas presented in this section. However, in the next sections
we will apply our theory to soliton hierarchies and then all the expressions
appearing in this section will become convergent (in fact, finite); the
$\mathcal{A}$-valued functions $\mathbb{K}_{n}$ will then become Frobenius
integrable deformations of soliton hierarchies.

######  Definition 1.

We say that the set (finite or not)
$\left\\{\mathbb{K}_{0},\mathbb{K}_{1},\ldots\right\\}$ of
$\mathcal{A}$-valued functions
$\mathbb{K}_{i}=\mathbb{K}_{i}(t_{0},\ldots,t_{i})$ satisfies the Frobenius
integrability condition (in a triangular form) if

$\frac{\partial\mathbb{K}_{j}}{\partial
t_{i}}+\operatorname{ad}_{\mathbb{K}_{i}}\mathbb{K}_{j}=0,\qquad 0\leqslant
i<j.$ (2.2)

Here and in what follows $\operatorname{ad}_{\mathbb{K}_{i}}$ is the adjoint
action in the Lie algebra $\mathcal{A}$ so that
$\operatorname{ad}_{\mathbb{K}_{i}}\mathbb{K}_{j}\equiv\left[\mathbb{K}_{i},\mathbb{K}_{j}\right]$.
Note that the _triangular_ dependence of $\mathbb{K}_{n}$ on $t_{i}$ given in
(2.1) means that (2.2) is the actual complete Frobenius condition

$\frac{\partial\mathbb{K}_{j}}{\partial
t_{i}}-\frac{\partial\mathbb{K}_{i}}{\partial
t_{j}}+\left[\mathbb{K}_{i},\mathbb{K}_{j}\right]=0,\qquad 0\leqslant i<j,$

simply because (2.1) implies that $\frac{\partial\mathbb{K}_{i}}{\partial
t_{j}}=0$ for $i<j$.

We begin our exposition by presenting a well-known fact.

######  Proposition 2.

Consider the following linear initial value problem

$\frac{d\bm{v}}{dt}+\mathbb{A}\bm{v}=0,\qquad\bm{v}(0)=\bm{v}_{0},$ (2.3)

where $\mathbb{A}=\mathbb{A}(t)$ is some time-dependent linear operator acting
in $\mathcal{A}$, $\bm{v}=\bm{v}(t)$ is an $\mathcal{A}$-valued function and
where $\bm{v}_{0}\in\mathcal{A}$. Then the formal solution of this problem is

$\displaystyle\bm{v}(t)$
$\displaystyle=\bm{v}_{0}-\partial_{t}^{-1}\mathbb{A}\bm{v}_{0}+\partial_{t}^{-1}\mathbb{A}\partial_{t}^{-1}\mathbb{A}\bm{v}_{0}-\partial_{t}^{-1}\mathbb{A}\partial_{t}^{-1}\mathbb{A}\partial_{t}^{-1}\mathbb{A}\bm{v}_{0}+\ldots$
$\displaystyle\equiv(1+\partial_{t}^{-1}\mathbb{A})^{-1}\bm{v}_{0},$

where

$\partial_{t}^{-1}=\int_{0}^{t}dt^{\prime}$

is the formal linear operator of definite integration with respect to the time
variable $t$ so that $\partial_{t}^{-1}\mathbb{A}$ is a $t$-dependent operator
on $\mathcal{A}$.

###### Proof.

Indeed, given that for any linear operator $\mathbb{B}$ in $\mathcal{A}$ we
have, formally,

$(1+\mathbb{B})^{-1}\equiv
1-\mathbb{B}+\mathbb{B}^{2}-\mathbb{B}^{3}+\ldots=1-\mathbb{B}\,(1+\mathbb{B})^{-1},$
(2.4)

then, taking $\mathbb{B}=\partial_{t}^{-1}\mathbb{A}$, we obtain

$\displaystyle\frac{d\bm{v}(t)}{dt}$
$\displaystyle=\partial_{t}(1+\partial_{t}^{-1}\mathbb{A})^{-1}\bm{v}_{0}=\partial_{t}(\bm{v}_{0}-\partial_{t}^{-1}\mathbb{A}(1+\partial_{t}^{-1}\mathbb{A})^{-1}\bm{v}_{0})$
$\displaystyle=-\mathbb{A}(1+\partial_{t}^{-1}\mathbb{A})^{-1}\bm{v}_{0}=-\mathbb{A}\bm{v}(t).$

∎

In the case $\mathbb{A}$ does not depend on $t$ we obtain the well known
solution of the IVP (2.3) in the form of the exponential of $\mathbb{A}$:

$\frac{d\mathbb{A}}{dt}=0\quad\Rightarrow\quad\bm{v}(t)=(1+\partial_{t}^{-1}\mathbb{A})^{-1}\bm{v}_{0}=\exp(-t\mathbb{A})\bm{v}_{0}.$
(2.5)

We will now generalize this result to the case of a system of linear time-
dependent equations. As notified above, we will work with $\mathcal{A}$-valued
functions of some parameters $t_{i}$, which we will often call _times_.

######  Theorem 3.

Suppose that $n$ $\mathcal{A}$-valued functions
$\mathbb{K}_{0},\ldots,\mathbb{K}_{n-1}$ satisfy the following conditions:

$\displaystyle\mathbb{K}_{i}=\mathbb{K}_{i}(t_{0},\ldots,t_{i}),$
$\displaystyle\qquad i=0,1,\ldots,n-1,$ (2.6a)
$\displaystyle\frac{\partial\mathbb{K}_{j}}{\partial
t_{i}}+\operatorname{ad}_{\mathbb{K}_{i}}\mathbb{K}_{j}$
$\displaystyle=0,\qquad 0\leqslant i<j<n.$ (2.6b)

Then the initial value problem

$\displaystyle\frac{\partial\mathbb{K}}{\partial
t_{i}}+\operatorname{ad}_{\mathbb{K}_{i}}\mathbb{K}$ $\displaystyle=0,\qquad
i=0,\ldots,n-1,$ (2.7a) $\displaystyle\mathbb{K}(0,\ldots,0)$
$\displaystyle=\bar{\mathbb{K}}\in\mathcal{A}.$ (2.7b)

for the $\mathcal{A}$-valued function
$\mathbb{K}=\mathbb{K}(t_{0},\ldots,t_{n-1})$ has the formal solution

$\mathbb{K}(t_{0},\ldots,t_{n-1})=(1+\partial_{t_{n-1}}^{-1}\operatorname{ad}_{\mathbb{K}_{n-1}})^{-1}\cdots(1+\partial_{t_{1}}^{-1}\operatorname{ad}_{\mathbb{K}_{1}})^{-1}(1+\partial_{t_{0}}^{-1}\operatorname{ad}_{\mathbb{K}_{0}})^{-1}\bar{\mathbb{K}},$
(2.8)

where

$\partial_{t_{i}}^{-1}f(t_{i})\equiv\int_{0}^{t_{i}}f(t)\,dt.$

Assuming that $\bar{\mathbb{K}}$ in (2.7b) depends on an additional evolution
parameter $t_{n}$ we arrive at the following corollary.

######  Corollary 4.

Suppose that $n$ $\mathcal{A}$-valued functions
$\mathbb{K}_{0},\ldots,\mathbb{K}_{n-1}$ satisfy the conditions (2.6). Suppose
also that an $\mathcal{A}$-valued function $\mathbb{K}_{n}\mathbb{=}$
$\mathbb{K}_{n}(t_{0},\ldots,t_{n})$ satisfies the following initial value
problem

$\displaystyle\frac{\partial\mathbb{K}_{n}}{\partial
t_{i}}+\operatorname{ad}_{\mathbb{K}_{i}}\mathbb{K}_{n}$
$\displaystyle=0,\qquad 0\leqslant i<n,\qquad$ (2.9a)
$\displaystyle\mathbb{K}_{n}(\underset{n}{\underbrace{0,\ldots,0}},t_{n})$
$\displaystyle=\bar{\mathbb{K}}_{n}(t_{n}),$ (2.9b)

where $\bar{\mathbb{K}}_{n}(t_{n})$ is an $\mathcal{A}$-valued function of
$t_{n}$. Then, the IVP (2.9) has the unique (formal) solution

$\mathbb{K}_{n}(t_{0},\ldots,t_{n})=(1+\partial_{t_{n-1}}^{-1}\operatorname{ad}_{\mathbb{K}_{n-1}})^{-1}\cdots(1+\partial_{t_{1}}^{-1}\operatorname{ad}_{\mathbb{K}_{1}})^{-1}(1+\partial_{t_{0}}^{-1}\operatorname{ad}_{\mathbb{K}_{0}})^{-1}\bar{\mathbb{K}}_{n}(t_{n}).$
(2.10)

For the proof of Theorem 3, see Appendix A. Let us now comment this theorem
and the corollary that follows. The conditions (2.6) means that the
$\mathcal{A}$-valued functions $\mathbb{K}_{0},\ldots,\mathbb{K}_{n-1}$
satisfy the Frobenius integrability condition (2.2). Further in the article
the $\mathcal{A}$-valued functions $\mathbb{K}_{i}$ are vector fields on some
finite or infinite-dimensional manifold $\mathcal{M}$, so that
$\mathbb{K}_{i}=\mathbb{K}_{i}(t_{0},\ldots,t_{i},u)$ with $u\in\mathcal{M}$,
and the Frobenius condition (2.2) in turn implies that the corresponding non-
autonomous dynamical systems

$\frac{du}{dt_{i}}=\mathbb{K}_{i}(t_{0},\ldots,t_{i},u),\qquad
i=0,\ldots,n-1,$

posses (at least locally) a common, multi-time solution
$u=u(t_{0},\ldots,t_{n-1};u_{0})$ through each point $u_{0}$ of the manifold
$\mathcal{M}$. Theorem 3 yields us then a tool to add one more vector field
$\mathbb{K}$ to the set
$\left\\{\mathbb{K}_{0},\ldots,\mathbb{K}_{n-1}\right\\}$ so that the set
$\left\\{\mathbb{K}_{0},\ldots,\mathbb{K}_{n-1},\mathbb{K}_{n}\equiv\mathbb{K}\right\\}$
still satisfies the Frobenius integrability condition (2.2). If now the
initial condition $\bar{\mathbb{K}}$ for (2.7) depends additionally on a new
evolution parameter $t_{n}$ then the solution (2.10) in Corollary 4 provides
us with the set
$\left\\{\mathbb{K}_{0},\ldots,\mathbb{K}_{n-1},\mathbb{K}_{n}\equiv\mathbb{K}\right\\}$
of non-autonomous vector fields, depending now on one more evolution parameter
$t_{n}$, and such that they satisfy the Frobenius condition (2.2) for
$0\leqslant i<j\leqslant n$.

Thus, Corollary 4 is a useful tool to recursively construct sets of non-
autonomous vector fields, _triangularly_ depending on $t_{i}$ and satisfying
the Frobenius condition. This corollary will be used in the following sections
to produce non-autonomous Frobenius integrable deformations of various soliton
hierarchies.

Let us conclude this section with a specification of Theorem 3 to the
situation when $\mathbb{K}_{i}=\mathbb{K}_{i}(t_{0},\ldots,t_{i-1})$ only (in
such situation we say that neither of $\mathbb{K}_{i}$ depends on its own
evolution parameter $t_{i}$). In this particular situation, in accordance with
(2.5),

$(1+\partial_{t_{i}}^{-1}\operatorname{ad}_{\mathbb{K}_{i}})^{-1}=\exp(-t_{i}\operatorname{ad}_{\mathbb{K}_{i}}),$

and thus we have the following remark.

######  Remark 5.

If $\mathbb{K}_{i}=\mathbb{K}_{i}(t_{0},\ldots,t_{i-1})$ for $i=0,\ldots,n-1$
and $\bar{\mathbb{K}}_{n}(t_{n})=\bar{\mathbb{K}}_{n}$, i.e. if neither of
$\bar{\mathbb{K}}_{i}$ depends on its own evolution parameter $t_{i}$, then
the solution of the IVP (2.9) takes the form

$\mathbb{K}_{n}(t_{0},\ldots,t_{n-1})=\exp(-t_{n-1}\operatorname{ad}_{\mathbb{K}_{n-1}})\cdots\exp(-t_{1}\operatorname{ad}_{\mathbb{K}_{1}})\exp(-t_{0}\operatorname{ad}_{\mathbb{K}_{0}})\bar{\mathbb{K}}_{n}.$
(2.11)

Naturally, in this case

$\mathbb{K}_{n}(0,\ldots,0)=\bar{\mathbb{K}}_{n}.$

##  3\. Frobenius integrability in hereditary algebras

The formulas (2.10) and (2.11) cannot be of any practical use until we have
some method to compute the expressions on their right hand sides. To achieve
this we will assume that our non-abelian Lie algebra $\mathcal{A}$ is a semi-
product of a commutative algebra and the Witt algebra (a centerless Virasoro
symmetry algebra), i.e. $\mathcal{A}$ is the so-called hereditary algebra [1,
2, 3]. Specifically, we assume that the hereditary algebra $\mathcal{A}$ is
generated by the elements $K_{n}\in$ $\mathcal{A}$, $n=1,2,\ldots$, and
$\sigma_{m}\in$ $\mathcal{A}$, $m=-1,0,1,\ldots$, such that

$[K_{n},K_{m}]=0,\qquad[\mathcal{\sigma}_{n},K_{m}]=(\alpha
m+\rho^{\prime}-1)K_{n+m},\qquad[\sigma_{n},\sigma_{m}]=\alpha(m-n)\sigma_{n+m},$

where $\rho^{\prime},\alpha\in\mathbb{R}$ and $\alpha\neq 0$. This choice is
motivated by the fact that hereditary algebras of soliton hierarchies, that we
will consider in this article, have this structure. By a simple rescaling of
all $\sigma_{n}$ we can always set $\alpha=1$. Thus, in this article we will
consider the hereditary algebra $\mathcal{A}$ with the generators $K_{n}$ and
$\sigma_{m}$ satisfying the commutation relations

$\displaystyle[K_{n},K_{m}]$ $\displaystyle=0,\qquad m,n=1,2,\ldots,$
$\displaystyle{}[\mathcal{\sigma}_{n},K_{m}]$
$\displaystyle=\kappa_{m}K_{n+m},\qquad n=-1,0,1,\ldots,\quad m=1,2,\ldots,$
(3.1) $\displaystyle{}[\sigma_{n},\sigma_{m}]$
$\displaystyle=(m-n)\sigma_{n+m},\qquad m,n=-1,0,1,\ldots,$

(so that $\rho-1\equiv\frac{1}{\alpha}\left(\rho^{\prime}-1\right)$) and where
we denote $K_{0}\equiv 0$. Here and further on we use the notation

$\kappa_{m}\equiv\rho+m-1.$ (3.2)

We can now choose our initial conditions $\bar{\mathbb{K}}_{n}(t_{n})$ in
(2.9b) as some very particular deformations (times-dependent linear
combinations) of the above generators of $\mathcal{A}$. It turns out that
formulas (2.10) and (2.11) attain a compact, finite form in two particular
cases:

  1. (1)

$\bar{\mathbb{K}}_{n}(t_{n})$ is a $\mathcal{A}_{-1}$-valued function for all
$n$,

  2. (2)

$\bar{\mathbb{K}}_{n}(t_{n})$ is a $\mathcal{A}_{0}$-valued function for all
$n$,

where

$\mathcal{A}_{-1}:=\operatorname{span}\left\\{\sigma_{-1},K_{1},K_{2},\ldots\right\\}\quad\text{and$\quad$}\mathcal{A}_{0}:=\operatorname{span}\left\\{\sigma_{0},K_{1},K_{2},\ldots\right\\}$

are two particular subalgebras of the hereditary algebra $\mathcal{A}$. They
are exceptional in the sense that for any $n$ the sets
$\mathcal{A}_{\varepsilon}^{(n)}\equiv\operatorname{span}\left\\{\sigma_{i},K_{1},K_{2},\ldots,K_{n}\right\\}$,
where $\varepsilon=-1$ or $\varepsilon=0$, are finite dimensional subalgebras
of $\mathcal{A}$. On the level of the algebras $\mathcal{A}_{\varepsilon}$ we
can interpret the solutions $\mathbb{K}_{n}$ of IVP (2.9) as leading to
construction of new deformed bases of $\mathcal{A}_{\varepsilon}$ satisfying
the Frobenius integrability condition (2.2).

###  3.1. Frobenius integrability in the hereditary subalgebra
$\mathcal{A}_{-1}$

We begin with the first case, i.e. when $\bar{\mathbb{K}}_{n}(t_{n})$ is a
$\mathcal{A}_{-1}$-valued function.

######  Theorem 6.

Consider the IVP (2.9) with the initial conditions (2.9b) in the form

$\mathbb{K}_{n}(0,\ldots,0,t_{n})=\bar{\mathbb{K}}_{n}(t_{n})\equiv\sigma_{-1}+\sum_{i=1}^{n}\mathbf{a}_{n,i}\left(t_{n}\right)K_{i},\qquad
n=0,1,...,$ (3.3)

where $\mathbf{a}_{n,i}(t_{n})$ are arbitrary differentiable functions (thus
$\bar{\mathbb{K}}_{n}(t_{n})$ are $\mathcal{A}_{-1}$-valued functions). Then,
the solution (2.10) of the IVP (2.9) is unique and attains the form

$\mathbb{K}_{n}=\sigma_{-1}+\sum_{i=1}^{n}\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})K_{i},$
(3.4)

where

$\mathbf{u}_{n,1}=-\mathbf{c}_{n},\qquad\mathbf{u}_{n,i}=\frac{(-1)^{i}}{[\kappa_{i}]!}\partial_{t_{0}}^{i-1}\mathbf{c}_{n},\qquad
i=2,\ldots,n,$ (3.5)

with

$\begin{split}\mathbf{c}_{n}(t_{0},\ldots,t_{n})&\equiv\sum_{m=2}^{n-1}\sum_{r=2}^{m}\sum_{s=1}^{r-1}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-1)!}(\tau_{m-1})^{r-s-1}\left(\partial_{t_{m}}^{-1}\right)^{s}\mathbf{a}_{m,r}(t_{m})\\\
&\quad+\sum_{r=1}^{n}\frac{(-1)^{r}[\kappa_{r}]!}{(r-1)!}(\tau_{n-1})^{r-1}\mathbf{a}_{n,r}(t_{n}),\qquad
n\in\mathbb{N}.\end{split}$ (3.6)

Here and in what follows we use the shorthand notations:

$\tau_{m}=t_{0}+t_{1}+\ldots+t_{m},\qquad[\kappa_{r}]!=\kappa_{2}\kappa_{3}\cdots\kappa_{r-1}\kappa_{r},\quad
r>1,\qquad[\kappa_{1}]!=1,$ (3.7)

and

$\partial_{t_{m}}^{-1}f(t_{m})\equiv\int_{0}^{t_{m}}f(t)\,dt.$

Notice that, the first term in (3.6) disappears for $\mathbf{c}_{1}$ and
$\mathbf{c}_{2}$. Also notice that, after combining (3.5) with (3.6) and
simplifying, one can see that $\mathbf{u}_{n,i}$ are always polynomial in
variables $\kappa_{m}$. This means that there is no issue with division by
zero if one of (3.2), for some particular $\rho$, is equal to zero.

###### Proof.

Direct calculation yields

$\mathbf{c}_{n}(0,\ldots,0,t_{n})=-\mathbf{a}_{n,1}(t_{n}).$

Moreover,

$\displaystyle(\mathbf{c}_{n})_{t_{0}}$
$\displaystyle=\sum_{m=2}^{n-1}\sum_{r=3}^{m}\sum_{s=1}^{r-2}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{m-1})^{r-s-2}\left(\partial_{t_{m}}^{-1}\right)^{s}\mathbf{a}_{m,r}(t_{m})$
$\displaystyle\quad+\sum_{r=2}^{n}\frac{(-1)^{r}[\kappa_{r}]!}{(r-2)!}(\tau_{n-1})^{r-2}\mathbf{a}_{n,r}(t_{n}),$

so that

$\left(\partial_{t_{0}}\mathbf{c}_{n}\right)(0,\ldots,0,t_{n})=\kappa_{2}\,\mathbf{a}_{n,2}(t_{n}).$

Continuing differentiation of $\mathbf{c}_{n}$ with respect to $t_{0}$ we
obtain

$\left(\partial_{t_{0}}^{i-1}\mathbf{c}_{n}\right)(0,\ldots,0,t_{n})=(-1)^{i}[\kappa_{i}]!\,\mathbf{a}_{n,i}(t_{n}),\qquad
i=2,3,\ldots,n.$

That means that $\mathbb{K}_{n}(0,\ldots,0,t_{n})=\bar{\mathbb{K}}_{n}(t_{n})$
and thus (3.4) satisfies (for each $n$) the initial conditions (3.3). Further,
(3.4) satisfies (2.9a) if and only if

$\displaystyle(\mathbf{u}_{n,n})_{t_{j}}$ $\displaystyle=0,$ (3.8a)
$\displaystyle(\mathbf{u}_{n,i})_{t_{j}}+\kappa_{i}\mathbf{u}_{n,i+1}$
$\displaystyle=0,\qquad j\leqslant i\leqslant n-1,$ (3.8b)
$\displaystyle(\mathbf{u}_{n,i})_{t_{j}}+\kappa_{i}\left(\mathbf{u}_{n,i+1}-\mathbf{u}_{j,i+1}\right)$
$\displaystyle=0,\qquad 1\leqslant i\leqslant j-1,$ (3.8c)

see Appendix B with $\epsilon=-1$. In Appendix C it is showed that

$(\mathbf{c}_{n})_{t_{j}}\equiv(\mathbf{c}_{n})_{t_{0}}-(\mathbf{c}_{j})_{t_{0}},\qquad
1\leqslant j\leqslant n-1,$ (3.9)

and hence

$\partial_{t_{0}}^{i-1}(\mathbf{c}_{n})_{t_{j}}=\partial_{t_{0}}^{i}\mathbf{c}_{n}-\partial_{t_{0}}^{i}\mathbf{c}_{j},\qquad
1\leqslant i\leqslant n-1,$

while $\partial_{t_{0}}^{i}\mathbf{c}_{j}=0$ for $i\geqslant j$. These
properties and the fact that
$[\kappa_{i+1}]!\equiv[\kappa_{i}]!\,\kappa_{i+1}$ imply that the conditions
(3.8) are identically satisfied. So, all $\mathbb{K}_{n}$ given by (3.4)
satisfy the IVP (2.9) with (2.9b) being of the particular form (3.3). Since
the conditions (3.8), being linear equations, have a unique solution for any
initial conditions, the solutions (3.4) and (2.10) must – for the chosen
initial conditions – coincide. ∎

The first few non-autonomous vectors (3.4) for the general IVP (3.3) have the
form

$\displaystyle\mathbb{K}_{0}$ $\displaystyle=\sigma_{-1},$
$\displaystyle\mathbb{K}_{1}$
$\displaystyle=\sigma_{-1}+\mathbf{a}_{1,1}(t_{1})\,K_{1},$
$\displaystyle\mathbb{K}_{2}$
$\displaystyle=\sigma_{-1}+\Bigl{[}\mathbf{a}_{2,1}(t_{2})-\kappa_{2}\,\tau_{1}\,\mathbf{a}_{2,2}(t_{2})\Bigr{]}K_{1}+\mathbf{a}_{2,2}(t_{2})\,K_{2},$
$\displaystyle\mathbb{K}_{3}$
$\displaystyle=\sigma_{-1}+\left[\kappa_{2}\,\partial_{t_{2}}^{-1}\mathbf{a}_{2,2}(t_{2})+\mathbf{a}_{3,1}(t_{3})-\kappa_{2}\,\tau_{2}\,\mathbf{a}_{3,2}(t_{3})+\frac{1}{2}\kappa_{2}\kappa_{3}\,\tau_{2}^{2}\,\mathbf{a}_{3,3}(t_{3})\right]K_{1}$
$\displaystyle\quad+\Bigl{[}\mathbf{a}_{3,2}(t_{3})-\kappa_{3}\,\tau_{2}\,\mathbf{a}_{3,3}(t_{3})\Bigr{]}K_{2}+\mathbf{a}_{3,3}(t_{3})\,K_{3},$
$\displaystyle\mathbb{K}_{4}$
$\displaystyle=\sigma_{-1}+\biggl{[}\kappa_{2}\,\partial_{t_{2}}^{-1}\mathbf{a}_{2,2}(t_{2})+\kappa_{2}\,\partial_{t_{3}}^{-1}\mathbf{a}_{3,2}(t_{3})-\kappa_{2}\kappa_{3}\,\tau_{2}\,\partial_{t_{3}}^{-1}\mathbf{a}_{3,3}(t_{3})-\kappa_{2}\kappa_{3}\,\partial_{t_{3}}^{-2}\mathbf{a}_{3,3}(t_{3})$
$\displaystyle\qquad\qquad+\mathbf{a}_{4,1}(t_{4})-\kappa_{2}\,\tau_{3}\,\mathbf{a}_{4,2}(t_{4})+\frac{1}{2}\kappa_{2}\kappa_{3}\,\tau_{3}^{2}\,\mathbf{a}_{4,3}(t_{4})-\frac{1}{6}\kappa_{2}\kappa_{3}\kappa_{4}\,\tau_{3}^{3}\,\mathbf{a}_{4,4}(t_{4})\biggr{]}K_{1}$
$\displaystyle\quad+\left[\kappa_{3}\,\partial_{t_{3}}^{-1}\mathbf{a}_{3,3}(t_{3})+\mathbf{a}_{4,2}(t_{4})-\kappa_{3}\,\tau_{3}\,\mathbf{a}_{4,3}(t_{4})+\frac{1}{2}\kappa_{3}\kappa_{4}\,\tau_{3}^{2}\,\mathbf{a}_{4,4}(t_{4})\right]K_{2}$
$\displaystyle\quad+\Bigl{[}\mathbf{a}_{4,3}(t_{4})-\kappa_{4}\,\tau_{3}\,\mathbf{a}_{4,4}(t_{4})\Bigr{]}K_{3}+\mathbf{a}_{4,4}(t_{4})\,K_{4},$

where $\kappa_{m}$ are defined by (3.2) and $\tau_{m}$ by (3.7).

######  Example 7.

Suppose that the functions $\mathbf{a}_{n,i}\left(t_{n}\right)$ in the initial
conditions (3.3) are given by the simple choice
$\mathbf{a}_{n,i}\left(t_{n}\right)=0$ for $i=1,\ldots,n-1$, and
$\mathbf{a}_{n,n}\left(t_{n}\right)=t_{n}$. Then the initial conditions (3.3)
have the form

$\bar{\mathbb{K}}_{0}(t_{0})\equiv\sigma_{-1},\qquad\bar{\mathbb{K}}_{n}(t_{n})\equiv\sigma_{-1}+t_{n}\,K_{i},\qquad
n=1,2,...,$ (3.10)

and the solution of the IVP (2.9) is given by (3.4) and (3.5) with
$\mathbf{c}_{n}$ in the form

$\mathbf{c}_{n}(t_{0},\ldots,t_{n})=\sum_{m=2}^{n-1}\sum_{s=2}^{m}\frac{(-1)^{m-1}[\kappa_{m}]!}{(m-s)!s!}(\tau_{m-1})^{m-s}t_{m}^{s}+\frac{(-1)^{n}[\kappa_{n}]!}{(n-1)!}(\tau_{n-1})^{n-1}t_{n}.$
(3.11)

Then, using (3.5) with (3.11) we obtain the first $\mathbb{K}_{n}$ in (3.4) in
the explicit form

$\displaystyle\mathbb{K}_{0}$ $\displaystyle=\sigma_{-1},$
$\displaystyle\mathbb{K}_{1}$ $\displaystyle=\sigma_{-1}+t_{1}K_{1},$
$\displaystyle\mathbb{K}_{2}$
$\displaystyle=\sigma_{-1}-\kappa_{2}(t_{0}+t_{1})t_{2}K_{1}+t_{2}K_{2},$
$\displaystyle\mathbb{K}_{3}$
$\displaystyle=\sigma_{-1}+\frac{1}{2}\kappa_{2}\bigl{[}t_{2}^{2}+\kappa_{3}(t_{0}+t_{1}+t_{2})^{2}t_{3}\bigr{]}K_{1}-\kappa_{3}(t_{0}+t_{1}+t_{2})t_{3}K_{2}+t_{3}K_{3},$
$\displaystyle\mathbb{K}_{4}$
$\displaystyle=\sigma_{-1}+\frac{1}{2}\kappa_{2}\Bigl{[}t_{2}^{2}-\frac{1}{3}\kappa_{3}(3t_{0}+3t_{1}+3t_{2}+t_{3})t_{3}^{2}-\frac{1}{3}\kappa_{3}\kappa_{4}\left(t_{0}+t_{1}+t_{2}+t_{3}\right)^{3}t_{4}\Bigr{]}K_{1}$
$\displaystyle\quad+\frac{1}{2}\kappa_{3}\bigl{[}t_{3}^{2}+\kappa_{4}(t_{0}+t_{1}+t_{2}+t_{3})^{2}t_{4}\bigr{]}K_{2}-\kappa_{4}(t_{0}+t_{1}+t_{2}+t_{3})t_{4}K_{3}+t_{4}K_{4}.$

###  3.2. Frobenius integrability in the hereditary subalgebra
$\mathcal{A}_{0}$

The second possibility arises when the initial conditions are given by
$\mathcal{A}_{0}$-valued functions.

######  Theorem 8.

Consider the IVP (2.9) with the initial conditions (2.9b) in the form

$\mathbb{K}_{n}(0,\ldots,0,t_{n})=\bar{\mathbb{K}}_{n}(t_{n})\equiv\sigma_{0}+\sum_{i=1}^{n}\mathbf{a}_{n,i}\left(t_{n}\right)K_{i}\text{,$\qquad$}n=0,1,2,...,$
(3.12)

where $\mathbf{a}_{n,i}(t_{n})$ are arbitrary differentiable functions (thus
$\bar{\mathbb{K}}_{n}(t_{n})$ is an $\mathcal{A}_{0}$-valued function). Then,
the solution (2.10) is unique and attains the form

$\mathbb{K}_{n}=\sigma_{0}+\sum_{i=1}^{n}\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})K_{i},$
(3.13)

where

$\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})=\sum_{r=i}^{n-1}\mathbf{c}_{r,i}\left(t_{r}\right)e^{-\kappa_{i}\tau_{r-1}}+\mathbf{a}_{n,i}\left(t_{n}\right)e^{-\kappa_{i}\tau_{n-1}},$
(3.14)

$\tau_{m}$ are again given by (3.7) and where $\mathbf{c}_{r,i}(t_{r})$ are
functions that satisfy non-homogeneous linear IVPs

$\mathbf{c}_{r,i}^{\prime}(t_{r})+\kappa_{i}\mathbf{c}_{r,i}(t_{r})=\kappa_{i}\mathbf{a}_{r,i}(t_{r}),\qquad\mathbf{c}_{r,i}(0)=0.$
(3.15)

Note that the solution of (3.15) is

$\mathbf{c}_{r,i}(t_{r})=\kappa_{i}\,e^{-\kappa_{i}t_{r}}\int_{0}^{t_{r}}\mathbf{a}_{r,i}(t)e^{\kappa_{i}t}\,dt.$
(3.16)

In particular, for the choice $\mathbf{a}_{r,i}(t_{i})=t_{i}^{m}$,
$m=0,1,\ldots$,

$\mathbf{c}_{r,i}(t_{r})=m!\left(\sum_{k=0}^{m}\frac{(-\kappa_{i})^{k-m}}{k!}t_{r}^{k}-e^{-\kappa_{i}t_{r}}\right),\qquad\kappa_{i}\neq
0.$ (3.17)

###### Proof.

Fix $n\in\mathbb{N}$ and assume that the solution $\mathbb{K}_{r}$ given by
(3.13) satisfy the IVP (2.9) for all $r<n$. Clearly,
$\mathbb{K}_{n}(0,\ldots,0,t_{n})=\bar{\mathbb{K}}_{n}(t_{n})$ so (3.13)
satisfy (for each $n$) the initial conditions (3.12). Further, (3.13)
satisfies (2.9a) if and only if

$\displaystyle(\mathbf{u}_{n,i})_{t_{j}}+\kappa_{i}\mathbf{u}_{n,i}$
$\displaystyle=0,\qquad j+1\leqslant i\leqslant n,$ (3.18a)
$\displaystyle(\mathbf{u}_{n,i})_{t_{j}}+\kappa_{i}\left(\mathbf{u}_{n,i}-\mathbf{u}_{j,i}\right)$
$\displaystyle=0,\qquad 1\leqslant i\leqslant j.$ (3.18b)

For proof of (3.18) see Appendix B with $\epsilon=0$. Note that the equations
(3.18) are linear and thus posses solutions for all $t_{j}$. Consider (3.14),
then for $j+1\leqslant i\leqslant n$ we have

$\displaystyle(\mathbf{u}_{n,i})_{t_{j}}$
$\displaystyle=-\kappa_{i}\sum_{r=i}^{n-1}\mathbf{c}_{r,i}\left(t_{r}\right)e^{-\kappa_{i}\tau_{r-1}}-\kappa_{i}\mathbf{a}_{n,i}\left(t_{n}\right)e^{-\kappa_{i}\tau_{n-1}}$
$\displaystyle\equiv-\kappa_{i}\mathbf{u}_{n,i},$

so that (3.18a) is identically true, while for $1\leqslant i\leqslant j$

$\displaystyle(\mathbf{u}_{n,i})_{t_{j}}$
$\displaystyle=\mathbf{c}_{j,i}^{\prime}\left(t_{j}\right)e^{-\kappa_{i}\tau_{j-1}}-\kappa_{i}\sum_{r=j+1}^{n-1}\mathbf{c}_{r,i}\left(t_{r}\right)e^{-\kappa_{i}\tau_{r-1}}-\kappa_{i}\mathbf{a}_{n,i}\left(t_{n}\right)e^{-\kappa_{i}\tau_{n-1}}$

and thus

$(\mathbf{u}_{n,i})_{t_{j}}+\kappa_{i}\left(\mathbf{u}_{n,i}-\mathbf{u}_{j,i}\right)=\left(\mathbf{c}_{j,i}^{\prime}\left(t_{j}\right)+\kappa_{i}\left[\mathbf{c}_{j,i}\left(t_{j}\right)-\mathbf{a}_{j,i}\left(t_{j}\right)\right]\right)e^{-\kappa_{i}\tau_{j-1}},$

so that (3.18b) is satisfied provided that the differential equations (3.15)
hold. As result, all (3.13) satisfy the IVP (2.9) with (2.9b) being of the
particular form (3.12) and so, by uniqueness of solutions of (3.18) for any
initial conditions, the solutions (3.13) and (2.10) must – for the chosen
initial conditions – coincide. ∎

The first few non-autonomous vectors (3.13) for the general IVP (3.12) have
the form

$\displaystyle\mathbb{K}_{0}$ $\displaystyle=\sigma_{0},$
$\displaystyle\mathbb{K}_{1}$
$\displaystyle=\sigma_{0}+\mathbf{a}_{1,1}(t_{1})e^{-\kappa_{1}\tau_{0}}K_{1},$
$\displaystyle\mathbb{K}_{2}$
$\displaystyle=\sigma_{0}+\Bigl{[}\mathbf{c}_{1,1}(t_{1})e^{-\kappa_{1}\tau_{0}}+\mathbf{a}_{2,1}(t_{2})e^{-\kappa_{1}\tau_{1}}\Bigr{]}K_{1}+\mathbf{a}_{2,2}(t_{2})e^{-\kappa_{2}\tau_{1}}K_{2},$
$\displaystyle\mathbb{K}_{3}$
$\displaystyle=\sigma_{0}+\Bigl{[}\mathbf{c}_{1,1}(t_{1})e^{-\kappa_{1}\tau_{0}}+\mathbf{c}_{2,1}(t_{2})e^{-\kappa_{1}\tau_{1}}+\mathbf{a}_{3,1}(t_{3})e^{-\kappa_{1}\tau_{2}}\Bigr{]}K_{1}$
$\displaystyle\quad+\Bigl{[}e^{-\kappa_{2}\tau_{1}}\mathbf{c}_{2,2}(t_{2})+\mathbf{a}_{3,2}(t_{3})e^{-\kappa_{2}\tau_{2}}\Bigr{]}K_{2}+\mathbf{a}_{3,3}(t_{3})e^{-\kappa_{3}\tau_{2}}K_{3},$
$\displaystyle\mathbb{K}_{4}$
$\displaystyle=\sigma_{0}+\Bigl{[}\mathbf{c}_{1,1}(t_{1})e^{-\kappa_{1}\tau_{0}}+\mathbf{c}_{2,1}(t_{2})e^{-\kappa_{1}\tau_{1}}+\mathbf{c}_{3,1}(t_{3})e^{-\kappa_{1}\tau_{2}}+\mathbf{a}_{4,1}(t_{4})e^{-\kappa_{1}\tau_{3}}\Bigr{]}K_{1}$
$\displaystyle\quad+\Bigl{[}\mathbf{c}_{2,2}(t_{2})e^{-\kappa_{2}\tau_{1}}+\mathbf{c}_{3,2}(t_{3})e^{-\kappa_{2}\tau_{2}}+\mathbf{a}_{4,2}(t_{4})e^{-\kappa_{2}\tau_{3}}\Bigr{]}K_{2}$
$\displaystyle\quad+\Bigl{[}\mathbf{c}_{3,3}(t_{3})e^{-\kappa_{3}\tau_{2}}+\mathbf{a}_{4,3}(t_{4})e^{-\kappa_{3}\tau_{3}}\Bigr{]}K_{3}+\mathbf{a}_{4,4}(t_{4})e^{-\kappa_{4}\tau_{3}}K_{4},$

where $\kappa_{m}$ are defined by (3.2), $\tau_{m}$ by (3.7) and
$\mathbf{c}_{r,i}(t_{r})$ are given by (3.16).

######  Example 9.

Suppose that the functions $\mathbf{a}_{n,i}\left(t_{n}\right)$ in the initial
conditions (3.12) are given by $\mathbf{a}_{n,i}(t_{n})=0$ for
$i=1,\ldots,n-1$ and $\mathbf{a}_{n,n}(t_{n})=t_{n}$, so that

$\bar{\mathbb{K}}_{0}(t_{0})\equiv\sigma_{-1},\qquad\bar{\mathbb{K}}_{n}(t_{n})\equiv\sigma_{-1}+t_{n}\,K_{i},\qquad
n=1,2,...\,.$ (3.19)

In this case, by (3.16) or (3.17)

$\mathbf{c}_{n,i}(t_{n})=0,\qquad
i=1,\ldots,n-1,\qquad\bm{c}_{n,n}(t_{n})=t_{n}+\frac{1}{\kappa_{n}}\left(e^{-\kappa_{n}t_{n}}-1\right),\qquad\kappa_{n}\neq
0.$ (3.20)

Then, if $\kappa_{i}\neq 0$, the first $\mathbb{K}_{n}$ in (3.13) have the
form

$\displaystyle\mathbb{K}_{0}$ $\displaystyle=\sigma_{0},$
$\displaystyle\mathbb{K}_{1}$
$\displaystyle=\sigma_{0}+t_{1}e^{-\kappa_{1}t_{0}}K_{1},$
$\displaystyle\mathbb{K}_{2}$
$\displaystyle=\sigma_{0}+\Bigl{[}\Bigl{(}t_{1}-\frac{1}{\kappa_{1}}\Bigr{)}e^{-\kappa_{1}t_{0}}+\frac{1}{\kappa_{1}}e^{-\kappa_{1}(t_{0}+t_{1})}\Bigr{]}K_{1}+t_{2}e^{-\kappa_{2}(t_{0}+t_{1})}K_{2},$
$\displaystyle\mathbb{K}_{3}$
$\displaystyle=\sigma_{0}+\Bigl{[}\Bigl{(}t_{1}-\frac{1}{\kappa_{1}}\Bigr{)}e^{-\kappa_{1}t_{0}}+\frac{1}{\kappa_{1}}e^{-\kappa_{1}(t_{0}+t_{1})}\Bigr{]}K_{1}$
$\displaystyle\quad+\Bigl{[}\Bigl{(}t_{2}-\frac{1}{\kappa_{2}}\Bigr{)}e^{-\kappa_{2}(t_{0}+t_{1})}+\frac{1}{\kappa_{2}}e^{-\kappa_{2}(t_{0}+t_{1}+t_{2})}\Bigr{]}K_{2}+t_{3}e^{-\kappa_{3}(t_{0}+t_{1}+t_{2})}K_{3},$
$\displaystyle\mathbb{K}_{4}$
$\displaystyle=\sigma_{0}+\Bigl{[}\Bigl{(}t_{1}-\frac{1}{\kappa_{1}}\Bigr{)}e^{-\kappa_{1}t_{0}}+\frac{1}{\kappa_{1}}e^{-\kappa_{1}(t_{0}+t_{1})}\Bigr{]}K_{1}$
$\displaystyle\quad+\Bigl{[}\Bigl{(}t_{2}-\frac{1}{\kappa_{2}}\Bigr{)}e^{-\kappa_{2}(t_{0}+t_{1})}+\frac{1}{\kappa_{2}}e^{-\kappa_{2}(t_{0}+t_{1}+t_{2})}\Bigr{]}K_{2}$
$\displaystyle\quad+\Bigl{[}\Bigl{(}t_{3}-\frac{1}{\kappa_{3}}\Bigr{)}e^{-\kappa_{3}(t_{0}+t_{1}+t_{2})}+\frac{1}{\kappa_{3}}e^{-\kappa_{3}(t_{0}+t_{1}+t_{2}+t_{3})}\Bigr{]}K_{3}+t_{4}e^{-\kappa_{4}(t_{0}+t_{1}+t_{2}+t_{3})}K_{4}.$

##  4\. Non-autonomous soliton hierarchies and their deformed isospectral
zero-curvature representations

From now on we will assume that the algebraic objects like $K$, $\mathbb{K}$,
or $\sigma$ are vector fields on some infinite-dimensional manifold
$\mathcal{M}$ with the corresponding autonomous evolution equations
$u_{t}=K[u]$ and non-autonomous evolution equations
$u_{t}=\mathbb{K}\left(x,t,[u]\right)$, where the square bracket denotes the
dependence on $u$ and a finite number of derivatives of $u$ w.r.t. $x$ (so
$[u]$ denotes jet-coordinates on $\mathcal{M}$) and where
$u=(u_{1}(x),\ldots,u_{N}(x))^{T}$ denotes points on the manifold
$\mathcal{M}$.

Consider thus an infinite hierarchy of mutually commuting autonomous
evolutionary equations on $\mathcal{M}$ of the form

$u_{s_{n}}=K_{n}[u],\qquad n=1,2\ldots,$ (4.1)

as well as a hierarchy of non-commuting evolutionary equations on
$\mathcal{M}$:

$u_{\tau_{n}}=\sigma_{n}[u],\qquad n=-1,0,1,\ldots,$ (4.2)

such that the commutation relations (3.1) are valid. The members of the
hierarchy (4.2) are called master symmetries for (4.1).

In this section we obtain – under Assumption 10 and Assumption 11 – the
Frobenius integrable non-autonomous hierarchies $u_{t_{n}}=\mathbb{K}_{n}[u]$,
where $\mathbb{K}_{n}$ is of the form (3.4) or (3.13), from an appropriate
deformation of an isospectral zero-curvature representation of (4.1) by a non-
standard (see Remark 12) isospectral zero-curve representation of (4.2).

######  Assumption 10.

Suppose that the commuting hierarchy (4.1) can be obtained from the
isospectral linear problem

$\begin{cases}\Psi_{x}=L\Psi,\\\
\Psi_{s_{i}}=U_{i}\Psi,&i=1,2,\ldots,\end{cases}$ (4.3)

where $L=L(\lambda,u)$, $U_{i}=U_{i}(\lambda,[u])$ are some $2\times 2$
matrices depending on $[u]$ and the auxiliary variable $\lambda$, s.t.
$\lambda_{s_{i}}=0$ for all $i$.

The subscript $s_{i}$ denotes the total derivative with respect to the
evolution parameter $s_{i}$. The compatibility condition, that is the
condition for existence of a common multi-time solution
$\Psi(x,s_{1},s_{2},\ldots)$, for the problem (4.3) is

$\displaystyle(\Psi_{x})_{s_{i}}$ $\displaystyle=(\Psi_{s_{i}})_{x},\qquad
i=1,2,\ldots,$ (4.4a) $\displaystyle(\Psi_{s_{i}})_{s_{j}}$
$\displaystyle=(\Psi_{s_{j}})_{s_{i}},\qquad i,j=1,2\ldots\,.$ (4.4b)

The condition (4.4a) is equivalent to

$L_{s_{i}}=\left[U_{i},L\right]+\left(U_{i}\right)_{x}\equiv
L^{\prime}\left[K_{i}\right],\qquad i=1,2,\ldots\,.$ (4.5)

Throughout the whole article $\Omega^{\prime}\left[K\right]$ denotes the
directional derivative of the tensor field $\Omega$ along the vector field $K$
on $\mathcal{M}$. The identity in (4.5) is the consequence of Assumption 10,
while the condition (4.4b) is equivalent to

$\left(U_{i}\right)_{s_{j}}-\left(U_{j}\right)_{s_{i}}+[U_{i},U_{j}]=0,\qquad
i,j=1,2,\ldots.$ (4.6)

Thus, Assumption 10 means that (4.5) is equivalent with the corresponding
equation $u_{s_{i}}=K_{i}[u]$ in (4.1), i.e. (4.5) is an isospectral zero-
curvature representation for (4.1). It also means that
$\Psi_{s_{i}}=\mathcal{L}_{K_{i}}\Psi$ where $\mathcal{L}$ is the Lie
derivative on $\mathcal{M}$. Then, the equation (4.6) guarantees that all
$K_{i}$ commute, since

$(\Psi_{s_{i}})_{s_{j}}-(\Psi_{s_{j}})_{s_{i}}=\mathcal{L}_{K_{j}}\mathcal{L}_{K_{i}}\Psi-\mathcal{L}_{K_{i}}\mathcal{L}_{K_{j}}\Psi=\mathcal{L}_{[K_{j},K_{i}]}\Psi=\Psi^{\prime}\bigl{[}[K_{j},K_{i}]\bigr{]}=0.$

Note also that (4.6) can be written as

$U_{i}^{\prime}[K_{j}]-U_{j}^{\prime}[K_{i}]+[U_{i},U_{j}]=0,\qquad
i,j=1,2,\ldots\,.$ (4.7)

######  Assumption 11.

Suppose also that the (non-commuting) hierarchy (4.2) of master symmetries can
be obtained from the following _deformed_ linear isospectral problem

$\begin{cases}\Psi_{x}=L\Psi,\\\
\Psi_{\tau_{i}}=V_{i}\Psi-\lambda^{i+1}\Psi_{\lambda},&i=-1,0,1\ldots,\end{cases}$
(4.8)

where $L=L(\lambda,u)$ is the same $L$ as in (4.3) while
$V_{i}=V_{i}(\lambda,[u])$ are some matrices depending on $[u]$ and $\lambda$
such that $\lambda_{\tau_{i}}=0$.

In (4.8) $\Psi_{\lambda}\equiv\frac{\partial\Psi}{\partial\lambda}$.
Obviously, we cannot expect that (4.8) posses a common multi-time solution
$\Psi(x,\tau_{-1},\tau_{0},\tau_{1},\ldots)$. Instead, the Assumption 11 means
that (4.8) has, for each $i$, a solution $\Psi(x,\tau_{i})$ so that

$(\Psi_{x})_{\tau_{i}}=(\Psi_{\tau_{i}})_{x},\qquad i=-1,0,\ldots,$

which is equivalent to

$L_{\tau_{i}}=\left[V_{i},L\right]+\left(V_{i}\right)_{x}-\lambda^{i+1}L_{\lambda}\equiv
L^{\prime}\left[\sigma_{i}\right],\qquad i=-1,0,1,\ldots,$ (4.9)

and the identity in (4.9) is the consequence of Assumption 11. This assumption
means thus that each equation in (4.9) is equivalent with the corresponding
equation $u_{\tau_{i}}=\sigma_{i}[u]$ in (4.2). Since the fields $\sigma_{i}$
in (4.2) do not commute we clearly cannot expect that
$(\Psi_{\tau_{i}})_{\tau_{j}}=(\Psi_{\tau_{j}})_{\tau_{i}}$. Instead we have

$(\Psi_{\tau_{i}})_{\tau_{j}}-(\Psi_{\tau_{j}})_{\tau_{i}}=\Psi^{\prime}\bigl{[}[\sigma_{j},\sigma_{i}]\bigr{]}=(i-j)\Psi^{\prime}[\sigma_{i+j}]=(i-j)\Psi_{\tau_{i+j}},\qquad
i,j=-1,0,1,\ldots,$

which is equivalent to

$(V_{i})_{\tau_{j}}-(V_{j})_{\tau_{i}}+[V_{i},V_{j}]+\lambda^{j+1}(V_{i})_{\lambda}-\lambda^{i+1}(V_{j})_{\lambda}=(i-j)V_{i+j},\qquad
i,j=-1,0,\ldots,$

that is to

$V_{i}^{\prime}[\sigma_{j}]-V_{j}^{\prime}[\sigma_{i}]+[V_{i},V_{j}]+\lambda^{j+1}(V_{i})_{\lambda}-\lambda^{i+1}(V_{j})_{\lambda}=(i-j)V_{i+j},\qquad
i,j=-1,0,\ldots\,.$ (4.10)

######  Remark 12.

Usually in literature (see for example [3]) one constructs a zero-curvature
representation for (4.2), from the _non-isospectral_ problem

$\begin{cases}\Psi_{x}=L\Psi,\\\
\Psi_{\tau_{i}}=V_{i}\Psi,&i=1,2,\ldots,\end{cases}$

with $\lambda_{\tau_{i}}=\lambda^{i+1}$. The isospectral problem (4.8) is
however equivalent (in the sense that it leads to the same zero-curvature
equations (4.10)) with the above isospectral problem, while being better
adapted to our needs.

We will now construct an isospectral zero-curvature representation of the
hierarchies

$u_{t_{n}}=\mathbb{K}_{n}[u],$

with $\mathbb{K}_{n}$ given in (3.4) or in (3.13) by combining the isospectral
problems (4.3) and (4.8). Consider thus the _deformed_ isospectral linear
problem

$\begin{cases}\Psi_{x}=L\Psi,\\\
\Psi_{t_{n}}=W_{n}\Psi-\lambda^{\varepsilon+1}\Psi_{\lambda},&n=1,2,\ldots,\end{cases}$
(4.11)

with $\varepsilon=-1$ or $\varepsilon=0$ and where $\lambda_{t_{n}}=0$, with
$W_{n}=W_{n}(\lambda,[u])$ defined as

$W_{n}=V_{\varepsilon}+\sum_{i=1}^{n}\mathbf{v}_{n,i}(t_{0},\ldots,t_{n})U_{i},$
(4.12)

where $L,V_{\varepsilon},U_{i}$ are given as above in this section and where
$\mathbf{v}_{n,i}$ are so far undetermined functions of evolution parameters
$t_{0},\ldots,t_{n}$.

######  Theorem 13.

The compatibility condition $(\Psi_{x})_{t_{n}}=(\Psi_{t_{n}})_{x}$ for (4.11)
has the form

$L_{t_{n}}=\left[W_{n},L\right]+\left(W_{n}\right)_{x}-\lambda^{\varepsilon+1}L_{\lambda}\equiv
L^{\prime}\left[u_{t_{n}}\right],\qquad n=1,2,\ldots,$ (4.13)

where

$u_{t_{n}}=\mathbb{K}_{n}\equiv\sigma_{\varepsilon}+\sum_{i=1}^{n}\mathbf{v}_{n,i}(t_{0},\ldots,t_{n})K_{i}[u],\qquad
n=1,2,\ldots\,.$ (4.14)

The identity in (4.13) means that (4.13) is equivalent with the corresponding
equation $u_{t_{n}}=\mathbb{K}_{n}[u]$ in (4.14). Note that so far the vector
fields $\mathbb{K}_{n}[u]$ in (4.14) have nothing in common with
$\mathbb{K}_{n}$ in (3.4) or (3.13).

###### Proof.

Due to the form of $W_{n}$ we have

$\displaystyle\left[W_{n},L\right]+\left(W_{n}\right)_{x}-\lambda^{\varepsilon+1}L_{\lambda}$
$\displaystyle=\left[V_{\varepsilon},L\right]+\left(V_{\varepsilon}\right)_{x}-\lambda^{\varepsilon+1}L_{\lambda}+\sum_{i=1}^{n}\mathbf{v}_{n,i}\left(\left[U_{i},L\right]+\left(U_{i}\right)_{x}\right)$
$\displaystyle=L^{\prime}\left[\sigma_{\varepsilon}\right]+\sum_{i=1}^{n}\mathbf{v}_{n,i}L^{\prime}\left[K_{i}\right]\equiv
L^{\prime}\left[\mathbb{K}_{n}\right].$

∎

So far the functions $\mathbf{v}_{n,i}$ are undetermined. Let us now demand
that the compatibility conditions

$(\Psi_{t_{m}})_{t_{n}}=(\Psi_{t_{n}})_{t_{m}},\qquad m,n=1,2,\ldots,$ (4.15)

for (4.11) are satisfied. These conditions are equivalent with the Frobenius
integrability conditions (2.2), since

$\displaystyle(\Psi_{t_{m}})_{t_{n}}-(\Psi_{t_{n}})_{t_{m}}=\left(\Psi^{\prime}[\mathbb{K}_{m}]\right)_{t_{n}}-\left(\Psi^{\prime}[\mathbb{K}_{n}]\right)_{t_{m}}$
$\displaystyle=\Psi^{\prime\prime}[\mathbb{K}_{n};\mathbb{K}_{m}]+\Psi^{\prime}\left[\frac{\partial\mathbb{K}_{m}}{\partial
t_{n}}+\mathbb{K}_{m}^{\prime}[\mathbb{K}_{n}]\right]-\Psi^{\prime\prime}[\mathbb{K}_{m};\mathbb{K}_{n}]-\Psi^{\prime}\left[\frac{\partial\mathbb{K}_{n}}{\partial
t_{m}}+\mathbb{K}_{n}^{\prime}[\mathbb{K}_{m}]\right]$
$\displaystyle=\Psi^{\prime}\left[\frac{\partial\mathbb{K}_{m}}{\partial
t_{n}}-\frac{\partial\mathbb{K}_{n}}{\partial
t_{m}}+[\mathbb{K}_{n},\mathbb{K}_{m}]\right],\qquad m,n=1,2,\ldots,$

where
$\Psi^{\prime\prime}[\mathbb{K}_{n};\mathbb{K}_{m}]=\Psi^{\prime\prime}[\mathbb{K}_{m};\mathbb{K}_{n}]$
is the second directional derivative. On the other hand (4.15) hold if and
only if

$(W_{n})_{t_{m}}-(W_{m})_{t_{n}}+[W_{n},W_{m}]+\lambda^{\varepsilon+1}(W_{n})_{\lambda}-\lambda^{\varepsilon+1}(W_{m})_{\lambda}=0.$
(4.16)

Let us thus investigate the conditions under which (4.16) hold. Assuming
$1\leqslant m<n,$ we have that $(\mathbf{v}_{m,i})_{t_{n}}=0$ and then the
zero-curvature relations (4.16) reduce to

$\frac{\partial W_{n}}{\partial
t_{m}}+W_{n}^{\prime}[\mathbb{K}_{m}]-W_{m}^{\prime}[\mathbb{K}_{n}]+[W_{n},W_{m}]+\lambda^{\varepsilon+1}(W_{n})_{\lambda}-\lambda^{\varepsilon+1}(W_{m})_{\lambda}=0\text{,}\qquad
1\leqslant m<n.$ (4.17)

We can now prove the following theorem.

######  Theorem 14.

The zero curvature conditions (4.17) are equivalent with the set of equations
on the functions $\mathbf{v}_{n,i}$ that is exactly the same as the set of
equations (3.8) and (3.18), respectively for $\varepsilon=-1$ and
$\varepsilon=0$, on the functions $\mathbf{u}_{n,i}$.

The proof of this theorem can be found in Appendix D. This theorem means that
the functions $\mathbf{v}_{n,i}$ and $\mathbf{u}_{n,i}$ pairwise coincide so
that the deformed isospectral problem (4.11) leads exactly to the Frobenius
integrable hierarchy $u_{t_{n}}=\mathbb{K}_{n}[u]$ with $\mathbb{K}_{n}$ given
by (3.4) (for $\varepsilon=-1$) or by (3.13) (for $\varepsilon=0$). Hence,

$W_{n}\equiv
V_{\varepsilon}+\sum_{i=1}^{n}\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})U_{i}.$
(4.18)

######  Remark 15.

Thus, we obtain the same non-autonomous Frobenius integrable hierarchies of
PDE’s starting from the deformed spectral problem (4.11) and starting from the
non-autonomous deformations in the case of the subalgebras
$\mathcal{A}_{\varepsilon}$ of the hereditary algebra (3.1), that we consider
in Section 3.

###  4.1. Hamiltonian structure of non-autonomous soliton hierarchies

We will now focus on soliton hierarchies. Suppose we have an infinite
hierarchy of vector fields $K_{n}$ on $\mathcal{M}$ that are bi-Hamiltonian
with respect to two compatible Poisson structures $\pi_{0}$ and $\pi_{1}$

$K_{n}=\pi_{0}\delta H_{n}=\pi_{1}\delta H_{n-1},\qquad n=1,2,\ldots,$

with $\pi_{0}$ being invertible. Then the operator $N=\pi_{1}\pi_{0}^{-1}$ is
an operator with the vanishing Nijenhuis torsion so that for any vector field
$K$ on $\mathcal{M}$ we have

$\mathcal{L}_{NK}N=N\mathcal{L}_{K}N.$ (4.19)

Any operator satisfying (4.19) is called a hereditary operator. Then,

$K_{n}\equiv N^{n-1}K_{1},\qquad n=2,3,\ldots,$ (4.20)

and by the hereditary property (4.19)

$\mathcal{L}_{K_{n}}N=0,\qquad n=1,2,\ldots,$

and

$[K_{n},K_{m}]=0,\qquad n,m=1,2,\ldots\,.$

Define now the infinite sequence of $1$-forms

$\gamma_{n}\equiv\delta H_{n}=\bigl{(}N^{\dagger}\bigr{)}^{i}\gamma_{0},\qquad
n=0,1,\ldots\,,$ (4.21)

where $\gamma_{0}=\delta H_{0}$ and where $N^{\dagger}=\pi_{0}^{-1}\pi_{1}$.
By the same hereditary property of $N$ they are all closed and thus there
exists an infinite sequence of functionals $H_{n}=\int h_{n}dx$ such that
$\gamma_{n}=\delta H_{n}$. Define now the infinite sequence of Poisson
operators

$\pi_{k}=N^{k}\pi_{0},\qquad k=2,3,\ldots\,.$

The operators $\pi_{k}$ are pairwise compatible and usually non-local. Then,
it follows that the field $K_{n}$ is $(n+1)$-Hamiltonian

$K_{n}=\pi_{0}\delta H_{n}=\pi_{1}\delta H_{n-1}=\ldots=\pi_{n}\delta
H_{0}\text{,$\qquad$}n=1,2,\ldots\,.$

Consider also a scaling vector field $\sigma_{0}$ such that

$\begin{cases}\mathcal{L}_{\sigma_{0}}K_{1}=\rho K_{1},&\rho\in\mathbf{R,}\\\
\mathcal{L}_{\sigma_{0}}N=N,\end{cases}$

and define the infinite sequences of vector fields $\sigma_{n}$ on
$\mathcal{M}$ through

$\sigma_{n}=N^{n}\mathcal{\sigma}_{0},\qquad n=-1,0,1,\ldots\,.$ (4.22)

Then it can be shown, using the hereditary property (4.19), that the vector
fields (4.20) and (4.22) satisfy the commutation relations (3.1).

Finally, let us assume that there exist a vector field $\sigma_{-1}$ such that
$\sigma_{0}=N\sigma_{-1}$ and such that it is Hamiltonian with respect to
$\pi_{0}$

$\mathcal{\sigma}_{-1}=\pi_{0}\delta F.$

Then, all $\sigma_{n}$ are Hamiltonian with respect to the Poisson operator
$\pi_{n+1}$, that is

$\sigma_{n}=\pi_{n+1}\delta F,\qquad n=1,2,...,$ (4.23)

and it immediately follows that every non-autonomous vector field
$\mathbb{K}_{n}$ in (3.4) or in (3.13) is also Hamiltonian (but not bi-
Hamiltonian), as

$\displaystyle\mathbb{K}_{n}$
$\displaystyle\equiv\sigma_{\varepsilon}+\sum_{i=1}^{n}\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})K_{i}[u]=$
$\displaystyle=\pi_{\varepsilon+1}\delta\left(F+\sum_{i=1}^{n}\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})H_{i-\varepsilon-1}[u]\right).$

##  5\. Non-autonomous KdV hierarchy

In this and following sections we apply our theory, developed above, to three
well known soliton hierarchies: Korteveg-de Vries, dispersive water waves and
Ablowitz-Kaup-Newell-Segur. The all have the structure exactly as described in
subsection 4.1.

###  5.1. KdV hierarchy

As a first illustration of our theory, consider the KdV hierarchy. The KdV
equation

$u_{t}=\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}$

is a member of the bi-Hamiltonian chain of nonlinear PDE’s

$u_{t_{i}}=K_{i}[u]=\pi_{0}\delta H_{i}=\pi_{1}\delta H_{i-1},\qquad
i=1,2,...,$ (5.1)

with two compatible Poisson operators

$\pi_{0}=\partial_{x},\qquad\pi_{1}=\frac{1}{4}\partial_{x}^{3}+u\partial_{x}+\frac{1}{2}u_{x}.$

The hierarchy (5.1) is _autonomous_ in the sense that none of the vector
fields $K_{i}[u]$ of the hierarchy depends explicitly on the evolution
parameters $t_{j}$. The KdV hierarchy (5.1) can be generated by the recursion
operator and its adjoint

$N\equiv\pi_{1}\pi_{0}^{-1}=\frac{1}{4}\partial_{x}^{2}+u+\frac{1}{2}u_{x}\partial_{x}^{-1},\qquad
N^{\dagger}=\frac{1}{4}\partial_{x}^{2}+u-\frac{1}{2}\partial_{x}^{-1}u_{x},$

in the sense that (4.20) and (4.21) are valid. In particular, we find that the
first vector fields $K_{n}$ have the form

$\begin{split}K_{1}&=u_{x},\qquad
K_{2}=\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x},\\\
K_{3}&=\frac{1}{16}u_{5x}+\frac{5}{8}uu_{3x}+\frac{5}{4}u_{x}u_{xx}+\frac{15}{8}u^{2}u_{x},\\\
K_{4}&=\frac{1}{64}u_{7x}+\frac{7}{32}uu_{5x}+\frac{21}{32}u_{x}u_{4x}+\frac{35}{32}u_{xx}u_{3x}+\frac{35}{32}u_{x}^{3}+\frac{35}{8}uu_{x}u_{xx}+\frac{35}{32}u^{2}u_{3x}+\frac{35}{16}u^{3}u_{x},\end{split}$

the first conserved one-forms (cosymmetries) $\gamma_{n}\equiv\delta H_{n}$
are

$\displaystyle\gamma_{0}$
$\displaystyle=2,\qquad\gamma_{1}=u,\qquad\gamma_{2}=\frac{1}{4}u_{xx}+\frac{3}{4}u^{2},$
$\displaystyle\gamma_{3}$
$\displaystyle=\frac{1}{16}u_{4x}+\frac{5}{8}uu_{xx}+\frac{5}{16}u_{x}^{2}+\frac{5}{8}u^{3},$
$\displaystyle\gamma_{4}$
$\displaystyle=\frac{1}{64}u_{6x}+\frac{7}{32}uu_{4x}+\frac{7}{16}u_{x}u_{3x}+\frac{21}{64}u_{xx}^{2}+\frac{35}{32}u^{2}u_{xx}+\frac{35}{32}uu_{x}^{2}+\frac{35}{64}u^{4},$

while the first Hamiltonian densities $h_{n}$ of the conserved functionals
$H_{n}=\int h_{n}\,dx$ are

$\begin{split}h_{0}&=2u,\qquad h_{1}=\frac{1}{2}u^{2},\qquad
h_{2}=-\frac{1}{8}u_{x}^{2}+\frac{1}{4}u^{3},\\\
h_{3}&=\frac{1}{32}u_{xx}^{2}+\frac{5}{32}u^{2}u_{xx}+\frac{5}{32}u^{4},\\\
h_{4}&=-\frac{1}{128}u_{3x}^{2}+\frac{7}{64}uu_{xx}^{2}-\frac{35}{64}u^{2}u_{x}^{2}+\frac{7}{64}u^{5}.\end{split}$

With the KdV hierarchy (5.1) one can also relate the hierarchy of its master
symmetries (4.22) with the first few $\sigma_{n}$ of the form

$\displaystyle\sigma_{-1}$
$\displaystyle=1,\qquad\sigma_{0}=u+\frac{1}{2}xu_{x},$
$\displaystyle\sigma_{1}$
$\displaystyle=\frac{1}{2}u_{xx}+\frac{1}{8}xu_{3x}+u^{2}+\frac{1}{2}xuu_{x}+\frac{1}{4}u_{x}\partial_{x}^{-1}u.$

The master symmetries $\sigma_{n}$ are in general non-local. They are
Hamiltonian due to (4.23) with the conserved functional

$F=\int xu\,dx.$

The symmetries $K_{i}$ and master symmetries $\sigma_{j}$ of the KdV equation
generate the hereditary algebra (3.1) with

$\rho=\frac{1}{2}\qquad\text{so that}\qquad\kappa_{n}=n-\frac{1}{2}.$ (5.2)

The matrix Lax representation (4.5) for the KdV hierarchy is given by

$L=\begin{pmatrix}0&1\\\ \lambda-u&0\end{pmatrix},$ (5.3)

and

$U_{n}=\frac{1}{2}\sum_{i=0}^{n-1}\begin{pmatrix}-\frac{1}{2}\left(\gamma_{i}\right)_{x}&\gamma_{i}\\\
(\lambda-u)\gamma_{i}-\frac{1}{2}\left(\gamma_{i}\right)_{xx}&\frac{1}{2}\left(\gamma_{i}\right)_{x}\end{pmatrix}\lambda^{n-i-1},\qquad
n=1,2,\ldots\,.$

In particular, $U_{1}=L$,

$U_{2}=\begin{pmatrix}-\frac{1}{4}u_{x}&\lambda+\frac{1}{2}u\\\
\lambda^{2}-\frac{1}{2}u\lambda-\frac{1}{2}u^{2}-\frac{1}{4}u_{xx}&\frac{1}{4}u_{x}\end{pmatrix}$

and

$U_{3}=\begin{pmatrix}-\frac{1}{4}u_{x}\lambda-\frac{1}{16}(u_{3x}+6uu_{x})&\lambda^{2}+\frac{1}{2}u\lambda+\frac{1}{8}(u_{xx}+3u^{2})\\\
\lambda^{3}-\frac{1}{2}u\lambda^{2}-\frac{1}{8}(u_{xx}+u^{2})\lambda-\frac{1}{16}u_{4x}+\frac{1}{2}uu_{xx}+\frac{3}{8}u_{x}^{2}+\frac{3}{8}u^{3}&\frac{1}{4}u_{x}\lambda+\frac{1}{16}(u_{3x}+6uu_{x})\end{pmatrix}.$

The Lax formulation for the hierarchy of the KdV master symmetries
$\sigma_{n}$ is given by (4.9) with $V_{n}$ of the form

$V_{n}=\frac{1}{2}\sum_{i=-1}^{n-1}\begin{pmatrix}-\frac{1}{2}\sigma_{i}&\partial_{x}^{-1}\sigma_{i}\\\
(\lambda-u)\partial_{x}^{-1}\sigma_{i}-\frac{1}{2}\left(\sigma_{i}\right)_{x}&\frac{1}{2}\sigma_{i}\end{pmatrix}\lambda^{n-i-1},\qquad
n=-1,0,1\ldots,$

so that

$V_{-1}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},\qquad
V_{0}=\begin{pmatrix}-\frac{1}{4}&\frac{1}{2}x\\\
\frac{1}{2}(\lambda-u)x&\frac{1}{4}\end{pmatrix}.$

###  5.2. Non-autonomous KdV hierarchy in the case $\mathcal{A}_{-1}$

We now present the deformed KdV hierarchy $u_{t_{n}}=\mathbb{K}_{n}[u]$ for
the case of the hereditary subalgebra $\mathcal{A}_{-1}$.

For the general initial conditions (3.3) the first members (3.4) of the non-
autonomous KdV hierarchy take the form

$\displaystyle u_{t_{0}}$ $\displaystyle=1,$ $\displaystyle u_{t_{1}}$
$\displaystyle=1+\mathbf{a}_{1,1}(t_{1})\,u_{x},$ $\displaystyle u_{t_{2}}$
$\displaystyle=1+\Bigl{[}\mathbf{a}_{2,1}(t_{2})-\frac{3}{2}(t_{0}+t_{1})\,\mathbf{a}_{2,2}(t_{2})\Bigr{]}u_{x}+\mathbf{a}_{2,2}(t_{2})\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]},$
$\displaystyle u_{t_{3}}$
$\displaystyle=1+\left[\frac{3}{2}\,\partial_{t_{2}}^{-1}\mathbf{a}_{2,2}(t_{2})+\mathbf{a}_{3,1}(t_{3})-\frac{3}{2}(t_{0}+t_{1}+t_{2})\mathbf{a}_{3,2}(t_{3})+\frac{15}{8}(t_{0}+t_{1}+t_{2})^{2}\mathbf{a}_{3,3}(t_{3})\right]u_{x}$
$\displaystyle\quad+\Bigl{[}\mathbf{a}_{3,2}(t_{3})-\frac{5}{2}(t_{0}+t_{1}+t_{2})\mathbf{a}_{3,3}(t_{3})\Bigr{]}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]}$
$\displaystyle\quad+\mathbf{a}_{3,3}(t_{3})\Bigl{[}\frac{1}{16}u_{5x}+\frac{5}{8}uu_{3x}+\frac{5}{4}u_{x}u_{xx}+\frac{15}{8}u^{2}u_{x}\Bigr{]}.$

If we consider the initial conditions given by the choice
$\mathbf{a}_{n,i}(t_{n})=\delta_{i,n}t_{n}$, that is (3.10), the deformed
vector fields $\mathbb{K}_{n}$ are given by (3.4) with (3.5) specified by
(3.11) with $\rho$ and $\kappa_{m}$ as in (5.2). In this case the first few
non-autonomous vector fields $\mathbb{K}_{n}$ are given by formulas in Example
7 and so the first few members of our hierarchy specify to

$\displaystyle u_{t_{0}}$ $\displaystyle=1,$ $\displaystyle u_{t_{1}}$
$\displaystyle=1+t_{1}\,u_{x},$ $\displaystyle u_{t_{2}}$
$\displaystyle=1-\frac{3}{2}\,(t_{0}+t_{1})t_{2}\,u_{x}+t_{2}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]},$
$\displaystyle u_{t_{3}}$
$\displaystyle=1+\left[\frac{3}{4}t_{2}^{2}+\frac{15}{8}(t_{0}+t_{1}+t_{2})^{2}t_{3}\right]u_{x}-\frac{5}{2}(t_{0}+t_{1}+t_{2})t_{3}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]}$
$\displaystyle\quad+t_{3}\Bigl{[}\frac{1}{16}u_{5x}+\frac{5}{8}uu_{3x}+\frac{5}{4}u_{x}u_{xx}+\frac{15}{8}u^{2}u_{x}\Bigr{]}.$

The Lax representation of the above non-autonomous KdV hierarchy is given by
(4.13), with $L$ as in (5.3) and $W_{n}$ defined by (4.18) with
$\varepsilon=-1,$ where, in the general case, $\mathbf{u}_{n,i}$ are given by
(3.5) and in the case of special initial conditions (3.10) the functions
$\mathbf{u}_{n,i}$ are specified by (3.11).

###  5.3. Non-autonomous KdV hierarchy in the case $\mathcal{A}_{0}$

The deformed KdV hierarchy $u_{t_{n}}=\mathbb{K}_{n}[u]$ for the case of the
hereditary subalgebra $\mathcal{A}_{0}$ and with the general initial
conditions (3.12) is given by the vector fields $\mathbb{K}_{n}$ as in (3.13),
thus in particular

$\displaystyle u_{t_{0}}$ $\displaystyle=u+\frac{1}{2}xu_{x},$ $\displaystyle
u_{t_{1}}$
$\displaystyle=u+\frac{1}{2}xu_{x}+\mathbf{a}_{1,1}(t_{1})e^{-\frac{1}{2}t_{0}}u_{x},$
$\displaystyle u_{t_{2}}$
$\displaystyle=u+\frac{1}{2}xu_{x}+\Bigl{[}\mathbf{c}_{1,1}(t_{1})e^{-\frac{1}{2}t_{0}}+\mathbf{a}_{2,1}(t_{2})e^{-\frac{1}{2}(t_{0}+t_{1})}\Bigr{]}u_{x}+\mathbf{a}_{2,2}(t_{2})e^{-\frac{3}{2}(t_{0}+t_{1})}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]},$
$\displaystyle u_{t_{3}}$
$\displaystyle=u+\frac{1}{2}xu_{x}+\Bigl{[}\mathbf{c}_{1,1}(t_{1})e^{-\frac{1}{2}t_{0}}+\mathbf{c}_{2,1}(t_{2})e^{-\frac{1}{2}(t_{0}+t_{1})}+\mathbf{a}_{3,1}(t_{3})e^{-\frac{1}{2}(t_{0}+t_{1}+t_{2})}\Bigr{]}u_{x}$
$\displaystyle\quad+\Bigl{[}e^{-\frac{3}{2}(t_{0}+t_{1})}\mathbf{c}_{2,2}(t_{2})+\mathbf{a}_{3,2}(t_{3})e^{-\frac{3}{2}(t_{0}+t_{1}+t_{2})}\Bigr{]}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]}$
$\displaystyle\quad+\mathbf{a}_{3,3}(t_{3})e^{-\frac{3}{2}(t_{0}+t_{1}+t_{2})}\Bigl{[}\frac{1}{64}u_{7x}+\frac{7}{32}uu_{5x}+\frac{21}{32}u_{x}u_{4x}+\frac{35}{32}u_{xx}u_{3x}+\frac{35}{32}u_{x}^{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\frac{35}{8}uu_{x}u_{xx}+\frac{35}{32}u^{2}u_{3x}+\frac{35}{16}u^{3}u_{x}\Bigr{]}.$

If we consider the initial conditions (3.19) the deformed vector fields
$\mathbb{K}_{n}$ are given by (3.13) with (3.14) specified by (3.20) with
$\rho$ and $\kappa_{m}$ given by (5.2). In this case, as in Example 9, we have

$\displaystyle u_{t_{0}}$ $\displaystyle=u+\frac{1}{2}xu_{x},$ $\displaystyle
u_{t_{1}}$
$\displaystyle=u+\frac{1}{2}xu_{x}+t_{1}e^{-\frac{1}{2}t_{0}}u_{x},$
$\displaystyle u_{t_{2}}$
$\displaystyle=u+\frac{1}{2}xu_{x}+\left(t_{1}+2e^{-\frac{1}{2}t_{1}}-2\right)e^{-\frac{1}{2}t_{0}}u_{x}+t_{2}e^{-\frac{3}{2}(t_{0}+t_{1})}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]},$
$\displaystyle u_{t_{3}}$
$\displaystyle=u+\frac{1}{2}xu_{x}+\left(t_{1}+2e^{-\frac{1}{2}t_{1}}-2\right)e^{-\frac{1}{2}t_{0}}u_{x}+\left(t_{2}+\frac{2}{3}e^{-\frac{3}{2}t_{2}}-\frac{2}{3}\right)e^{-\frac{3}{2}(t_{0}+t_{1})}\Bigl{[}\frac{1}{4}u_{xxx}+\frac{3}{2}uu_{x}\Bigr{]}$
$\displaystyle\quad+t_{3}e^{-\frac{3}{2}(t_{0}+t_{1}+t_{2})}\Bigl{[}\frac{1}{64}u_{7x}+\frac{7}{32}uu_{5x}+\frac{21}{32}u_{x}u_{4x}+\frac{35}{32}u_{xx}u_{3x}+\frac{35}{32}u_{x}^{3}+\frac{35}{8}uu_{x}u_{xx}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{35}{32}u^{2}u_{3x}+\frac{35}{16}u^{3}u_{x}\Bigr{]}.$

The Lax representation of the above non-autonomous KdV hierarchies is given by
(4.13), with $L$ given by (5.3) and with respective $W_{n}$ defined by (4.18)
with $\varepsilon=0$.

##  6\. Non-autonomous DWW hierarchy

###  6.1. Autonomous DWW hierarchy

Let us now apply our theory to the DWW hierarchy. Its is a bi-Hamiltonian
hierarchy given by

$\begin{pmatrix}u\\\
v\end{pmatrix}_{t_{n}}=K_{n}=\pi_{0}\gamma_{n}=\pi_{1}\gamma_{n-1},\qquad
n=1,2,...,$ (6.1)

where $\gamma_{n}\equiv\delta H_{n}$ are exact one-forms and where

$\pi_{0}=\begin{pmatrix}-\frac{1}{2}v\partial_{x}-\frac{1}{2}\partial_{x}v&\partial_{x}\\\
\partial_{x}&0\end{pmatrix},\qquad\pi_{1}=\begin{pmatrix}\frac{1}{4}\partial_{x}^{3}+\frac{1}{2}u\partial_{x}+\frac{1}{2}\partial_{x}u&0\\\
0&\partial_{x}\end{pmatrix}$

are two compatible Poisson operators. This hierarchy is generated by (4.20)
and (4.21) with the recursion operator and its adjoint given by

$N=\pi_{1}\pi_{0}^{-1}=\begin{pmatrix}0&\frac{1}{4}\partial_{x}^{2}+u+\frac{1}{2}u_{x}\partial_{x}^{-1}\\\
1&v+\frac{1}{2}v_{x}\partial_{x}^{-1}\end{pmatrix},\qquad
N^{{\dagger}}=\begin{pmatrix}0&1\\\
\frac{1}{4}\partial_{x}^{2}+u-\frac{1}{2}\partial_{x}^{-1}u_{x}&v-\frac{1}{2}\partial_{x}^{-1}v_{x}\end{pmatrix}.$

In particular, we have the symmetries

$\displaystyle K_{1}$ $\displaystyle=\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix},\qquad
K_{2}=\begin{pmatrix}\frac{1}{4}v_{xxx}+uv_{x}+\frac{1}{2}vu_{x}\\\
u_{x}+\frac{3}{2}vv_{x}\end{pmatrix},$ $\displaystyle K_{3}$
$\displaystyle=\begin{pmatrix}\frac{3}{8}v^{2}u_{x}+\frac{3}{2}uvv_{x}+\frac{3}{2}uu_{x}+\frac{1}{4}u_{3x}+\frac{3}{8}vv_{3x}+\frac{9}{8}v_{x}v_{2x}\\\
\frac{3}{2}vu_{x}+\frac{3}{2}uv_{x}+\frac{15}{8}v^{2}v_{x}+\frac{1}{4}v_{3x}\end{pmatrix},$

cosymmetries

$\gamma_{0}=\begin{pmatrix}2\\\
v\end{pmatrix},\qquad\gamma_{1}=\begin{pmatrix}v\\\
u+\frac{3}{4}v^{2}\end{pmatrix},\qquad\gamma_{2}=\begin{pmatrix}u+\frac{3}{4}v^{2}\\\
\frac{1}{4}v_{xx}+\frac{3}{2}uv+\frac{5}{8}v^{3}\end{pmatrix},$

and functionals $H_{n}=\int h_{n}\,dx$, where

$h_{0}=2u+\frac{1}{2}v^{2},\qquad h_{1}=uv+\frac{1}{4}v^{3},\qquad
h_{2}=-\frac{1}{8}v_{x}^{2}+\frac{1}{2}u^{2}+\frac{3}{4}uv^{2}+\frac{5}{32}v^{4}.$

With the DWW hierarchy (6.1) one can also relate the hierarchy of its master
symmetries (4.22) with the first few $\sigma_{n}$ of the form

$\sigma_{-1}=\begin{pmatrix}-v\\\
2\end{pmatrix},\qquad\sigma_{0}=\begin{pmatrix}2u+xu_{x}\\\
v+xv_{x}\end{pmatrix},\qquad\sigma_{1}=\begin{pmatrix}\frac{3}{4}v_{xx}+\frac{1}{4}xv_{xxx}+uv+xuv_{x}+\frac{1}{2}xvu_{x}\\\
xu_{x}+\frac{3}{2}xvv_{x}+v^{2}+2u\end{pmatrix}.$

The master symmetries $\sigma_{n}$ are Hamiltonian as in (4.23) with

$F=\int\left(2xu+\frac{1}{2}xv^{2}\right)\,dx.$

The symmetries $K_{i}$ and the master symmetries $\sigma_{j}$ constitute the
hereditary algebra (3.1) with

$\rho=1\qquad\text{so that}\qquad\kappa_{n}=n.$ (6.2)

The zero-curvature formulation (4.5) for the DWW hierarchy is given by

$L=\begin{pmatrix}0&1\\\ \lambda^{2}-v\lambda-u&0\end{pmatrix}$ (6.3)

and by

$U_{n}=\frac{1}{2}\sum_{i=0}^{n-1}\begin{pmatrix}-\frac{1}{2}\left(\gamma_{i1}\right)_{x}&\gamma_{i1}\\\
\left(\lambda^{2}-v\lambda-u\right)\gamma_{i1}-\frac{1}{2}\left(\gamma_{i1}\right)_{xx}&\frac{1}{2}\left(\gamma_{i1}\right)_{x}\end{pmatrix}\lambda^{n-i-1},\qquad
n=1,2,\ldots,$ (6.4)

with $\gamma_{i1}$ denoting the first component of $\gamma_{i}$. In
particular, $U_{1}=L$ and

$U_{2}=\begin{pmatrix}-\frac{1}{4}v_{x}&\lambda+\frac{1}{2}v\\\
\lambda^{3}-\frac{1}{2}v\lambda^{2}-\left(u+\frac{1}{2}v^{2}\right)\lambda-\frac{1}{2}uv-\frac{1}{4}v_{2x}&\frac{1}{4}v_{x}\end{pmatrix},$
$U_{3}=\begin{pmatrix}-\frac{1}{4}v_{x}\lambda-\frac{1}{4}u_{x}-\frac{3}{8}vv_{x}&\lambda^{2}+\frac{1}{2}v\lambda+\frac{3}{8}v^{2}+\frac{1}{2}u\\\
(U_{3})_{21}&\frac{1}{4}v_{x}\lambda+\frac{1}{4}u_{x}+\frac{3}{8}vv_{x}\end{pmatrix},$

where

$(U_{3})_{21}=\lambda^{4}-\frac{1}{2}v\lambda^{3}-\frac{1}{8}\left(4u+v^{2}\right)\lambda^{2}-\frac{1}{8}\left(8uv+3v^{3}+2v_{2x}\right)\lambda-\frac{1}{8}\left(4u^{2}+3uv^{2}+2u_{2x}+3v_{x}^{2}+3vv_{2x}\right).$

The deformed Lax formulation for the hierarchy of the DWW master symmetries
$\sigma_{n}$ is given by (4.9) with $V_{n}$ of the form

$V_{n}=\frac{1}{2}\sum_{i=-1}^{n-1}\begin{pmatrix}-\frac{1}{2}\sigma_{i2}&\partial_{x}^{-1}\sigma_{i2}\\\
\left(\lambda^{2}-v\lambda-u\right)\partial_{x}^{-1}\sigma_{i2}-\frac{1}{2}\left(\sigma_{i2}\right)_{x}&\frac{1}{2}\sigma_{i2}\end{pmatrix}\lambda^{n-i-1},\qquad
n=-1,0,\ldots\,,$

with $\sigma_{i2}$ denoting the second component of $\sigma_{i}$, thus

$V_{-1}=0,\qquad V_{0}=\begin{pmatrix}-\frac{1}{2}&x\\\
\left(\lambda^{2}-v\lambda-u\right)x&\frac{1}{2}\end{pmatrix}.$ (6.5)

###  6.2. Non-autonomous DWW hierarchy in the case $\mathcal{A}_{-1}$

The first few vector fields of the deformed DWW hierarchy
$u_{t_{n}}=\mathbb{K}_{n}[u]$ in the case $\mathcal{A}_{-1}$ and with the
general initial conditions (3.3) have the form

$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{0}}$
$\displaystyle=\begin{pmatrix}-v\\\ 2\end{pmatrix},$
$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{1}}$
$\displaystyle=\begin{pmatrix}-v\\\
2\end{pmatrix}+\mathbf{a}_{1,1}(t_{1})\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix},$ $\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{2}}$
$\displaystyle=\begin{pmatrix}-v\\\
2\end{pmatrix}+\Bigl{[}\mathbf{a}_{2,1}(t_{2})-2(t_{0}+t_{1})\,\mathbf{a}_{2,2}(t_{2})\Bigr{]}\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix}+\mathbf{a}_{2,2}(t_{2})\begin{pmatrix}\frac{1}{4}v_{xxx}+uv_{x}+\frac{1}{2}vu_{x}\\\
u_{x}+\frac{3}{2}vv_{x}\end{pmatrix}.$

If we consider the initial conditions (3.10) the deformed vector fields
$\mathbb{K}_{n}$ are given by (3.4) with (3.5) specified by (3.11) with $\rho$
and $\kappa_{n}$ as in (6.2). Explicitely,

$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{0}}$
$\displaystyle=\begin{pmatrix}-v\\\ 2\end{pmatrix},$
$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{1}}$
$\displaystyle=\begin{pmatrix}-v\\\
2\end{pmatrix}+t_{1}\begin{pmatrix}u_{x}\\\ v_{x}\end{pmatrix},$
$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{2}}$
$\displaystyle=\begin{pmatrix}-v\\\
2\end{pmatrix}-2(t_{0}+t_{1})t_{2}\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix}+t_{2}\begin{pmatrix}\frac{1}{4}v_{xxx}+uv_{x}+\frac{1}{2}vu_{x}\\\
u_{x}+\frac{3}{2}vv_{x}\end{pmatrix}.$

###  6.3. Non-autonomous DWW hierarchy in the case $\mathcal{A}_{0}$

The deformed DWW hierarchy $u_{t_{n}}=\mathbb{K}_{n}[u]$ in the case
$\mathcal{A}_{0}$ and with the general initial conditions (3.12) is given by
the vector fields (3.13), thus in particular

$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{0}}$
$\displaystyle=\begin{pmatrix}2u+xu_{x}\\\ v+xv_{x}\end{pmatrix},$
$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{1}}$
$\displaystyle=\begin{pmatrix}2u+xu_{x}\\\
v+xv_{x}\end{pmatrix}+\mathbf{a}_{1,1}(t_{1})e^{-t_{0}}\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix},$ $\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{2}}$
$\displaystyle=\begin{pmatrix}2u+xu_{x}\\\
v+xv_{x}\end{pmatrix}+\Bigl{[}\mathbf{c}_{1,1}(t_{1})e^{-t_{0}}+\mathbf{a}_{2,1}(t_{2})e^{-(t_{0}+t_{1})}\Bigr{]}\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix}$
$\displaystyle\quad+\mathbf{a}_{2,2}(t_{2})e^{-2(t_{0}+t_{1})}\begin{pmatrix}\frac{1}{4}v_{xxx}+uv_{x}+\frac{1}{2}vu_{x}\\\
u_{x}+\frac{3}{2}vv_{x}\end{pmatrix}.$

For the initial conditions (3.19) from Example 9 the deformed vector fields
$\mathbb{K}_{n}$ are given by (3.13) with (3.14) specified by (3.20) with
$\rho$ and $\kappa_{n}$ as in (6.2). In this case

$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{0}}$
$\displaystyle=\begin{pmatrix}2u+xu_{x}\\\ v+xv_{x}\end{pmatrix},$
$\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{1}}$
$\displaystyle=\begin{pmatrix}2u+xu_{x}\\\
v+xv_{x}\end{pmatrix}+t_{1}e^{-t_{0}}\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix},$ $\displaystyle\begin{pmatrix}u\\\ v\end{pmatrix}_{t_{2}}$
$\displaystyle=\begin{pmatrix}2u+xu_{x}\\\
v+xv_{x}\end{pmatrix}+\bigl{(}t_{1}+(e^{-t_{1}}-1)\bigr{)}e^{-t_{0}}\begin{pmatrix}u_{x}\\\
v_{x}\end{pmatrix}+t_{2}e^{-2(t_{0}+t_{1})}\begin{pmatrix}\frac{1}{4}v_{xxx}+uv_{x}+\frac{1}{2}vu_{x}\\\
u_{x}+\frac{3}{2}vv_{x}\end{pmatrix}.$

The Lax representation for all the above non-autonomous DWW hierarchies is
given by (4.13) with $L$ as in (6.3) and with $W_{n}$ as in (4.18) with
respective $\mathbf{u}_{n,i}$ and $\varepsilon=-1$ or $0$.

##  7\. Non-autonomous AKNS hierarchy

###  7.1. Autonomous AKNS hierarchy

In the last example we apply our theory to the AKNS hierarchy. It is a bi-
Hamiltonian hierarchy given by

$\begin{pmatrix}q\\\
r\end{pmatrix}_{t_{n}}=K_{n}=\pi_{0}\gamma_{n}=\pi_{1}\gamma_{n-1},\qquad
n=1,2,...,$

where $\gamma_{n}=\delta H_{n}$ are exact one-forms and where

$\pi_{0}=\begin{pmatrix}0&i\\\
-i&0\end{pmatrix},\qquad\pi_{1}=\begin{pmatrix}-q\partial_{x}^{-1}q&-\frac{1}{2}\partial_{x}+q\partial_{x}^{-1}r\\\
-\frac{1}{2}\partial_{x}+r\partial_{x}^{-1}q&-r\partial_{x}^{-1}r\end{pmatrix}$

are two compatible Poisson operators ($i^{2}=-1$ throughout this whole
section). This hierarchy is generated by (4.20) and (4.21) with the recursion
operator and its adjoint given by

$N=\pi_{1}\pi_{0}^{-1}=i\begin{pmatrix}\frac{1}{2}\partial_{x}-q\partial_{x}^{-1}r&-q\partial_{x}^{-1}q\\\
r\partial_{x}^{-1}r&-\frac{1}{2}\partial_{x}+r\partial_{x}^{-1}q\end{pmatrix},\qquad
N^{{\dagger}}=-i\begin{pmatrix}\frac{1}{2}\partial_{x}-r\partial_{x}^{-1}q&r\partial_{x}^{-1}r\\\
-q\partial_{x}^{-1}q&-\frac{1}{2}\partial_{x}+q\partial_{x}^{-1}r\end{pmatrix}.$

In particular, we have the symmetries

$K_{1}=i\begin{pmatrix}-2q\\\ 2r\end{pmatrix},\qquad
K_{2}=\begin{pmatrix}q_{x}\\\ r_{x}\end{pmatrix},\qquad
K_{3}=i\begin{pmatrix}\frac{1}{2}q_{xx}-q^{2}r\\\
-\frac{1}{2}r_{xx}+qr^{2}\end{pmatrix},\qquad
K_{4}=\begin{pmatrix}-\frac{1}{4}q_{xxx}+\frac{3}{2}qrq_{x}\\\
-\frac{1}{4}r_{xxx}+\frac{3}{2}qrr_{x}\end{pmatrix},$

cosymmetries

$\gamma_{1}=\begin{pmatrix}-2r\\\
-2q\end{pmatrix},\qquad\gamma_{2}=i\begin{pmatrix}r_{x}\\\
-q_{x}\end{pmatrix},\qquad\gamma_{3}=\begin{pmatrix}\frac{1}{2}r_{xx}-qr^{2}\\\
\frac{1}{2}q_{xx}-rq^{2}\end{pmatrix},\qquad\gamma_{4}=i\begin{pmatrix}-\frac{1}{4}r_{xxx}+\frac{3}{2}qrr_{x}\\\
\frac{1}{4}q_{xxx}-\frac{3}{2}qrq_{x}\end{pmatrix},$

and respective Hamiltonians $H_{n}=\int h_{n}\,dx$, where

$h_{1}=-2rq,\qquad h_{2}=-i\,rq_{x},\qquad
h_{3}=-\frac{1}{2}q_{x}r_{x}-\frac{1}{2}q^{2}r^{2},\qquad
h_{4}=i\left(\frac{1}{4}rq_{xxx}+\frac{3}{4}q^{2}rr_{x}\right).$

With the AKNS hierarchy (6.1) one can relate the hierarchy of its master
symmetries (4.22) with $\sigma_{-1}$ and $\sigma_{0}$ given by

$\sigma_{-1}=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix},\qquad\sigma_{0}=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}.$

One can directly check that in this case

$\rho=0\qquad\text{and}\qquad\kappa_{n}=n-1$ (7.1)

in the hereditary algebra (3.1). Moreover, the master symmetries $\sigma_{n}$
are Hamiltonian as in (4.23) with

$F=-2\int xqr\,dx.$

The zero-curvature formulation (4.5) for the AKNS hierarchy is given by [14,
15]

$L=\begin{pmatrix}-i\lambda&q\\\ r&i\lambda\end{pmatrix}$ (7.2)

and by

$U_{n}=i\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\lambda^{n-1}+\frac{i}{2}\sum_{j=1}^{n-1}\begin{pmatrix}-\partial_{x}^{-1}\left[rK_{j1}+qK_{j2}\right]&K_{j1}\\\
-K_{j2}&\partial_{x}^{-1}\left[rK_{j1}+qK_{j2}\right]\end{pmatrix}\lambda^{n-j-1},\qquad
n=1,2,\ldots\,.\text{ }$ (7.3)

In particular,

$\displaystyle U_{1}$ $\displaystyle=\begin{pmatrix}-i&0\\\
0&i\end{pmatrix},\qquad U_{2}\equiv L,\qquad
U_{2}=\begin{pmatrix}-i\lambda^{2}-\frac{i}{2}qr&q\lambda+\frac{i}{2}q_{x}\\\
r\lambda-\frac{i}{2}r_{x}&i\lambda^{2}+\frac{i}{2}qr\end{pmatrix},$
$\displaystyle U_{3}$
$\displaystyle=\begin{pmatrix}-i\lambda^{3}-\frac{i}{2}qr\lambda+\frac{1}{4}rq_{x}-\frac{1}{4}qr_{x}&q\lambda^{2}+\frac{i}{2}q_{x}\lambda-\frac{1}{4}q_{xx}+\frac{1}{2}q^{2}r\\\
r\lambda^{2}-\frac{i}{2}r_{x}\lambda-\frac{1}{4}r_{xx}+\frac{1}{2}qr^{2}&i\lambda^{3}+\frac{i}{2}qr\lambda-\frac{1}{4}rq_{x}+\frac{1}{4}qr_{x}\end{pmatrix}.$

The deformed Lax formulation for the hierarchy of the master symmetries
$\sigma_{n}$ is given by (4.9) with $V_{i}$ of the form

$V_{n}=i\begin{pmatrix}-x&0\\\
0&x\end{pmatrix}\lambda^{n+1}+\frac{i}{2}\sum_{j=-1}^{n-1}\begin{pmatrix}-\partial_{x}^{-1}\left[r\sigma_{j1}+q\sigma_{j2}\right]&\sigma_{j1}\\\
-\sigma_{j2}&\partial_{x}^{-1}\left[r\sigma_{j1}+q\sigma_{j2}\right]\end{pmatrix}\lambda^{n-j-1},$

where $n=-1,0,1,...$, and thus

$V_{-1}=\begin{pmatrix}-ix&0\\\ 0&ix\end{pmatrix},\qquad
V_{0}=\begin{pmatrix}-ix\lambda&xq\\\ xr&ix\lambda\end{pmatrix}.$ (7.4)

###  7.2. Non-autonomous AKNS hierarchy in the case $\mathcal{A}_{-1}$

The first four systems of the deformed AKNS hierarchy
$u_{t_{n}}=\mathbb{K}_{n}[u]$ in the case $\mathcal{A}_{-1}$ and with the
general initial conditions (3.3) are

$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{0}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\ 2xr\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{1}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix}+i\mathbf{\,a}_{1,1}(t_{1})\,\begin{pmatrix}-2q\\\
2r\end{pmatrix},$ $\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{2}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix}+i\Bigl{[}\mathbf{a}_{2,1}(t_{2})-(t_{0}+t_{1})\,\mathbf{a}_{2,2}(t_{2})\Bigr{]}\begin{pmatrix}-2q\\\
2r\end{pmatrix}+\mathbf{a}_{2,2}(t_{2})\,\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix},$ $\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{3}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix}+i\left[\partial_{t_{2}}^{-1}\mathbf{a}_{2,2}(t_{2})+\mathbf{a}_{3,1}(t_{3})-\tau_{2}\,\mathbf{a}_{3,2}(t_{3})+\tau_{2}^{2}\,\mathbf{a}_{3,3}(t_{3})\right]\begin{pmatrix}-2q\\\
2r\end{pmatrix}$
$\displaystyle\quad+\Bigl{[}\mathbf{a}_{3,2}(t_{3})-2\tau_{2}\,\mathbf{a}_{3,3}(t_{3})\Bigr{]}\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix}+i\mathbf{\,a}_{3,3}(t_{3})\,\begin{pmatrix}\frac{1}{2}q_{xx}-q^{2}r\\\
-\frac{1}{2}r_{xx}+qr^{2}\end{pmatrix},$

where $\tau_{2}=t_{0}+t_{1}+t_{2}$.

If we consider the initial conditions (3.10) the deformed vector fields
$\mathbb{K}_{n}$ are given by (3.4) with (3.5) specified by (3.11) with $\rho$
and $\kappa_{n}$ as in (7.1), yielding

$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{0}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\ 2xr\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{1}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix}+i\,t_{1}\,\begin{pmatrix}-2q\\\ 2r\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{2}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix}-i(t_{0}+t_{1})\,t_{2}\,\begin{pmatrix}-2q\\\
2r\end{pmatrix}+t_{2}\,\begin{pmatrix}q_{x}\\\ r_{x}\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{3}}$
$\displaystyle=i\begin{pmatrix}-2xq\\\
2xr\end{pmatrix}+i\left[\frac{1}{2}t_{2}^{2}+(t_{0}+t_{1}+t_{2})^{2}\,t_{3}\right]\begin{pmatrix}-2q\\\
2r\end{pmatrix}-2(t_{0}+t_{1}+t_{2})\,t_{3}\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix}$
$\displaystyle\quad+i\,t_{3}\,\begin{pmatrix}\frac{1}{2}q_{xx}-q^{2}r\\\
-\frac{1}{2}r_{xx}+qr^{2}\end{pmatrix}.$

###  7.3. Non-autonomous AKNS hierarchy in the case $\mathcal{A}_{0}$

The deformed AKNS hierarchy $u_{t_{n}}=\mathbb{K}_{n}[u]$ in the case
$\mathcal{A}_{0}$ and with the general initial conditions (3.12) is given by
the vector fields (3.13); the first few of them read

$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{0}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\ (xr)_{x}\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{1}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}+i\,\mathbf{a}_{1,1}(t_{1})\begin{pmatrix}-2q\\\
2r\end{pmatrix},$ $\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{2}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}+i\,\mathbf{a}_{2,1}(t_{2})\begin{pmatrix}-2q\\\
2r\end{pmatrix}+\mathbf{a}_{2,2}(t_{2})e^{-(t_{0}+t_{1})}\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix},$ $\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{3}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}+i\,\mathbf{a}_{3,1}(t_{3})\begin{pmatrix}-2q\\\
2r\end{pmatrix}$
$\displaystyle\quad+\Bigl{[}e^{-(t_{0}+t_{1})}\mathbf{c}_{2,2}(t_{2})+\mathbf{a}_{3,2}(t_{3})e^{-\tau_{2}}\Bigr{]}\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix}+i\,\mathbf{a}_{3,3}(t_{3})e^{-2\tau_{2}}\begin{pmatrix}\frac{1}{2}q_{xx}-q^{2}r\\\
-\frac{1}{2}r_{xx}+qr^{2}\end{pmatrix},$

where $\tau_{2}=t_{0}+t_{1}+t_{2}$. Notice that, in this case $\kappa_{1}=0$
and thus all $\mathbf{c}_{r,1}=0$ due to (3.16).

For the initial conditions (3.19) the deformed vector fields $\mathbb{K}_{n}$
are given by (3.13) with (3.14) specified by (3.20) with $\rho$ and
$\kappa_{n}$ as in (7.1), which yields

$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{0}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\ (xr)_{x}\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{1}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}+i\,t_{1}\begin{pmatrix}-2q\\\ 2r\end{pmatrix},$
$\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{2}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}+i\Bigl{[}\Bigl{(}t_{1}-\frac{1}{\kappa_{1}}\Bigr{)}e^{-\kappa_{1}t_{0}}+\frac{1}{\kappa_{1}}e^{-\kappa_{1}(t_{0}+t_{1})}\Bigr{]}\begin{pmatrix}-2q\\\
2r\end{pmatrix}+t_{2}e^{-\kappa_{2}(t_{0}+t_{1})}\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix},$ $\displaystyle\begin{pmatrix}q\\\ r\end{pmatrix}_{t_{3}}$
$\displaystyle=\begin{pmatrix}(xq)_{x}\\\
(xr)_{x}\end{pmatrix}+i\Bigl{[}\Bigl{(}t_{1}-\frac{1}{\kappa_{1}}\Bigr{)}e^{-\kappa_{1}t_{0}}+\frac{1}{\kappa_{1}}e^{-\kappa_{1}(t_{0}+t_{1})}\Bigr{]}\begin{pmatrix}-2q\\\
2r\end{pmatrix},$
$\displaystyle\quad+\Bigl{[}\Bigl{(}t_{2}-\frac{1}{\kappa_{2}}\Bigr{)}e^{-\kappa_{2}(t_{0}+t_{1})}+\frac{1}{\kappa_{2}}e^{-\kappa_{2}(t_{0}+t_{1}+t_{2})}\Bigr{]}\begin{pmatrix}q_{x}\\\
r_{x}\end{pmatrix}+i\,t_{3}e^{-\kappa_{3}(t_{0}+t_{1}+t_{2})}\begin{pmatrix}\frac{1}{2}q_{xx}-q^{2}r\\\
-\frac{1}{2}r_{xx}+qr^{2}\end{pmatrix}.$

The Lax representation for all the above non-autonomous AKNS hierarchies are
given by (4.13) with $L$ as in (6.3) and with $W_{n}$ as in (4.18) with
respective $\mathbf{u}_{n,i}$ and $\varepsilon=-1$ or $0$.

##  Appendix A

In this appendix we prove Theorem 3. Consider the following linear integral
operators

$\xi_{i}=1+\partial_{t_{i}}^{-1}\operatorname{ad}_{\mathbb{K}_{i}},\qquad
i=0,1,\ldots,n-1$

acting in $\mathcal{A}$ (notice that $\xi_{i}=\xi_{i}(t_{0},\ldots,t_{i})$ due
to (2.6a)). Then, by (2.4),

$\xi_{i}^{-1}=1-\partial_{t_{i}}^{-1}\operatorname{ad}_{\mathbb{K}_{i}}\xi_{i}^{-1}.$

Therefore

$\displaystyle\partial_{t_{i}}\xi_{i}$
$\displaystyle=\partial_{t_{i}}+\operatorname{ad}_{\mathbb{K}_{i}},$
$\displaystyle\partial_{t_{i}}\xi_{i}^{-1}$
$\displaystyle=\partial_{t_{i}}-\operatorname{ad}_{\mathbb{K}_{i}}\xi_{i}^{-1},$
(A.1)

and

$\partial_{t_{i}}\xi_{j}=\xi_{j}\partial_{t_{i}}\quad\text{for}\quad j<i,$

while

$\partial_{t_{i}}\xi_{j}^{-1}=-\xi_{j}^{-1}\frac{\partial\xi_{j}}{\partial
t_{i}}\xi_{j}^{-1}+\xi_{j}^{-1}\partial_{t_{i}}=-\xi_{j}^{-1}[\xi_{j},\operatorname{ad}_{\mathbb{K}_{i}}]\xi_{j}^{-1}+\xi_{j}^{-1}\partial_{t_{i}}\quad\text{for}\quad
j>i.$

Note that $\partial_{t_{i}}\xi_{j}$ and $\frac{\partial\xi_{j}}{\partial
t_{i}}$ are two different operators, as
$\partial_{t_{i}}\xi_{j}=\frac{\partial\xi_{j}}{\partial
t_{i}}+\xi_{j}\partial_{t_{i}}$. The formula above is due to the fact that for
any time-dependent operator $\xi_{j}$

$\frac{\partial\xi_{j}^{-1}}{\partial
t_{i}}=-\xi_{j}^{-1}\frac{\partial\xi_{j}}{\partial t_{i}}\xi_{j}^{-1}$ (A.2)

and the fact that the explicit time dependence of $\xi_{j}$ on $t_{i}$ is

$\displaystyle\frac{\partial\xi_{j}}{\partial t_{i}}$
$\displaystyle=\partial_{t_{j}}^{-1}\frac{\partial\operatorname{ad}_{\mathbb{K}_{j}}}{\partial
t_{i}}\overset{\eqref{w1}}{=}-\partial_{t_{j}}^{-1}\operatorname{ad}_{[\mathbb{K}_{i},\mathbb{K}_{j}]}=-\partial_{t_{j}}^{-1}[\operatorname{ad}_{\mathbb{K}_{i}},\operatorname{ad}_{\mathbb{K}_{j}}]=[\partial_{t_{j}}^{-1}\operatorname{ad}_{\mathbb{K}_{j}},\operatorname{ad}_{\mathbb{K}_{i}}]$
(A.3)
$\displaystyle=[\xi_{j},\operatorname{ad}_{\mathbb{K}_{i}}]\quad\text{for}\quad
i<j.$

Notice that $\frac{\partial\xi_{j}}{\partial t_{i}}=0$ for $i>j$.

Denote now $\bm{t}=(t_{0},\ldots,t_{n-1})$ and consider (2.8), that is

$\mathbb{K}\left(\bm{t}\right)=\xi_{n-1}^{-1}\cdots\xi_{1}^{-1}\xi_{0}^{-1}\bar{\mathbb{K}}.$

We will now show that $\mathbb{K}\left(\bm{t}\right)$ is the solution of the
IVP (2.7). Naturally, $\mathbb{K}\left(\bm{0}\right)=\bar{\mathbb{K}}$. Using
(A.2) we obtain

$\displaystyle\frac{\partial\mathbb{K}(\bm{t})}{\partial t_{i}}$
$\displaystyle=\partial_{t_{i}}\xi_{n-1}^{-1}\cdots\xi_{1}^{-1}\xi_{0}^{-1}\bar{\mathbb{K}}$
$\displaystyle=-\xi_{n-1}^{-1}\frac{\partial\xi_{n-1}}{\partial
t_{i}}\xi_{n-1}^{-1}\xi_{n-2}^{-1}\cdots\xi_{1}^{-1}\xi_{0}^{-1}\bar{\mathbb{K}}+\xi_{n-1}^{-1}\partial_{t_{i}}\xi_{n-2}^{-1}\cdots\xi_{1}^{-1}\xi_{0}^{-1}\bar{\mathbb{K}}=\cdots=$
$\displaystyle=-\sum_{j=i+1}^{n-1}\xi_{n-1}^{-1}\cdots\xi_{j}^{-1}\frac{\partial\xi_{j}}{\partial
t_{i}}\xi_{j}^{-1}\cdots\xi_{i+1}^{-1}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}+\xi_{n-1}^{-1}\cdots\xi_{i+1}^{-1}\partial_{t_{i}}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}$

and further by (A.3) and (A.1)

$\displaystyle\frac{\partial\mathbb{K}(\bm{t})}{\partial t_{i}}$
$\displaystyle=-\sum_{j=i+1}^{n-1}\xi_{n-1}^{-1}\cdots\xi_{j}^{-1}[\xi_{j},\operatorname{ad}_{\mathbb{K}_{i}}]\xi_{j}^{-1}\cdots\xi_{i+1}^{-1}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}-\xi_{n-1}^{-1}\cdots\xi_{i+1}^{-1}\operatorname{ad}_{\mathbb{K}_{i}}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}$
$\displaystyle=-\operatorname{ad}_{\mathbb{K}_{i}}\xi_{n-1}^{-1}\cdots\xi_{i+1}^{-1}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}-\sum_{j=i+1}^{n-2}\xi_{n-1}^{-1}\cdots\xi_{j+1}^{-1}\operatorname{ad}_{\mathbb{K}_{i}}\xi_{j}^{-1}\cdots\xi_{i+1}^{-1}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}$
$\displaystyle\quad+\sum_{j=i+1}^{n-1}\xi_{n-1}^{-1}\cdots\xi_{j}^{-1}\operatorname{ad}_{\mathbb{K}_{i}}\xi_{j-1}^{-1}\cdots\xi_{i}^{-1}\xi_{i-1}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}-\xi_{n-1}^{-1}\cdots\xi_{i+1}^{-1}\operatorname{ad}_{\mathbb{K}_{i}}\xi_{i}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}$
$\displaystyle=-\operatorname{ad}_{\mathbb{K}_{i}}\xi_{n-1}^{-1}\cdots\xi_{0}^{-1}\bar{\mathbb{K}}\equiv-\operatorname{ad}_{\mathbb{K}_{i}}\mathbb{K}(\bm{t}).$

##  Appendix B

We prove here (3.8) and (3.18) for the cases $\varepsilon=-1$ and
$\varepsilon=0$, respectively. In either case we have

$[\sigma_{\varepsilon},K_{i}]=\kappa_{i}K_{i+\varepsilon},\qquad i\geqslant
1,\quad K_{0}\equiv 0.$

Consider (3.13) and (3.4), thus

$\mathbb{K}_{n}=\sigma_{\varepsilon}+\sum_{i=1}^{n}\mathbf{u}_{n,i}(t_{0},\ldots,t_{n})K_{i},\qquad\varepsilon=0,-1.$

Then for $m<n$:

$\displaystyle\frac{\partial\mathbb{K}_{n}}{\partial
t_{m}}+\left[\mathbb{K}_{m},\mathbb{K}_{n}\right]=\sum_{i=1}^{n}(\mathbf{u}_{n,i})_{t_{m}}K_{i}+\sum_{i=1}^{n}\mathbf{u}_{n,i}\left[\sigma_{\varepsilon},K_{i}\right]+\sum_{j=1}^{m}\mathbf{u}_{m,j}\left[K_{j},\sigma_{\varepsilon}\right]$
$\displaystyle\qquad=\sum_{i=1}^{n}(\mathbf{u}_{n,i})_{t_{m}}K_{i}+\sum_{i=1-\varepsilon}^{n}\kappa_{i}\mathbf{u}_{n,i}K_{i+\varepsilon}-\sum_{i=1-\varepsilon}^{m}\kappa_{i}\mathbf{u}_{m,i}K_{i+\varepsilon}$
$\displaystyle\qquad=-\varepsilon(\mathbf{u}_{n,n})_{t_{m}}K_{n}+\sum_{i=m+1}^{n}\left[(\mathbf{u}_{n,i+\varepsilon})_{t_{m}}+\kappa_{i}\mathbf{u}_{n,i}\right]K_{i+\varepsilon}$
$\displaystyle\qquad\quad+\sum_{i=1-\varepsilon}^{m}\left[(\mathbf{u}_{n,i+\varepsilon})_{t_{m}}+\kappa_{i}(\mathbf{u}_{n,i}-\mathbf{u}_{m,i})\right]K_{i+\varepsilon}.$

Thus, $\frac{\partial\mathbb{K}_{n}}{\partial
t_{m}}+\left[\mathbb{K}_{m},\mathbb{K}_{n}\right]=0$ for $m<n$ if and only if

$\begin{cases}\varepsilon(\mathbf{u}_{n,n})_{t_{m}}=0,\\\
(\mathbf{u}_{n,i})_{t_{m}}+\kappa_{i-\varepsilon}\mathbf{u}_{n,i-\varepsilon}=0,&m+1+\varepsilon\leqslant
i\leqslant n+\varepsilon,\\\
(\mathbf{u}_{n,i})_{t_{m}}+\kappa_{i-\varepsilon}\left(\mathbf{u}_{n,i-\varepsilon}-\mathbf{u}_{m,i-\varepsilon}\right)=0,&1\leqslant
i\leqslant m+\varepsilon,\end{cases}$

which yields (3.18) for $\varepsilon=0$ and (3.8) for $\varepsilon=-1$.

##  Appendix C

We prove here (3.9). Differentiating (3.6) with respect to $t_{j}$ yields

$\displaystyle(\mathbf{c}_{n})_{t_{j}}$
$\displaystyle=\sum_{m=j+1}^{n-1}\sum_{r=3}^{m}\sum_{s=1}^{r-2}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{m-1})^{r-s-2}\left(\partial_{t_{m}}^{-1}\right)^{s}\mathbf{a}_{m,r}(t_{m})$
$\displaystyle\quad+\sum_{r=2}^{j}\sum_{s=1}^{r-1}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-1)!}(\tau_{j-1})^{r-s-1}\left(\partial_{t_{j}}^{-1}\right)^{s-1}\mathbf{a}_{j,r}(t_{j})$
$\displaystyle\quad+\sum_{r=2}^{n}\frac{(-1)^{r}[\kappa_{r}]!}{(r-2)!}(\tau_{n-1})^{r-2}\mathbf{a}_{n,r}(t_{n}),$

and further

$\displaystyle(\mathbf{c}_{n})_{t_{j}}$
$\displaystyle=\sum_{m=j}^{n-1}\sum_{r=3}^{m}\sum_{s=1}^{r-2}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{m-1})^{r-s-2}\left(\partial_{t_{m}}^{-1}\right)^{s}\mathbf{a}_{m,r}(t_{m})$
$\displaystyle\quad-\sum_{r=3}^{j}\sum_{s=1}^{r-2}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{j-1})^{r-s-2}\left(\partial_{t_{j}}^{-1}\right)^{s}\mathbf{a}_{j,r}(t_{j})$
$\displaystyle\quad+\sum_{r=2}^{j}\sum_{s=1}^{r-1}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-1)!}(\tau_{j-1})^{r-s-1}\left(\partial_{t_{j}}^{-1}\right)^{s-1}\mathbf{a}_{j,r}(t_{j})$
$\displaystyle\quad+\sum_{r=2}^{n}\frac{(-1)^{r}[\kappa_{r}]!}{(r-2)!}(\tau_{n-1})^{r-2}\mathbf{a}_{n,r}(t_{n}).$

Thus

$\displaystyle(\mathbf{c}_{n})_{t_{j}}$
$\displaystyle=\sum_{m=j}^{n-1}\sum_{r=3}^{m}\sum_{s=1}^{r-2}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{m-1})^{r-s-2}\left(\partial_{t_{m}}^{-1}\right)^{s}\mathbf{a}_{m,r}(t_{m})$
$\displaystyle\quad+\left(\sum_{r=2}^{j}\sum_{s=0}^{r-2}-\sum_{r=3}^{j}\sum_{s=1}^{r-2}\right)\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{j-1})^{r-s-2}\left(\partial_{t_{j}}^{-1}\right)^{s}\mathbf{a}_{j,r}(t_{j})$
$\displaystyle\quad+\sum_{r=2}^{n}\frac{(-1)^{r}[\kappa_{r}]!}{(r-2)!}(\tau_{n-1})^{r-2}\mathbf{a}_{n,r}(t_{n}),$

and finally

$\displaystyle(\mathbf{c}_{n})_{t_{j}}$
$\displaystyle=\left(\sum_{m=2}^{n-1}-\sum_{m=2}^{j-1}\right)\sum_{r=3}^{m}\sum_{s=1}^{r-2}\frac{(-1)^{r-1}[\kappa_{r}]!}{(r-s-2)!}(\tau_{m-1})^{r-s-2}\left(\partial_{t_{m}}^{-1}\right)^{s}\mathbf{a}_{m,r}(t_{m})$
$\displaystyle\quad+\left(\sum_{r=2}^{n}-\sum_{r=2}^{j}\right)\frac{(-1)^{r}[\kappa_{r}]!}{(r-2)!}(\tau_{n-1})^{r-2}\mathbf{a}_{n,r}(t_{n})$
$\displaystyle\equiv(\mathbf{c}_{n})_{t_{0}}-(\mathbf{c}_{j})_{t_{0}}.$

##  Appendix D

We prove here Theorem 14. We start by observing that, due to (3.1)

$(\Psi_{s_{i}})_{\tau_{j}}-(\Psi_{\tau_{j}})_{s_{i}}=\Psi^{\prime}\bigl{[}[\sigma_{j},K_{i}]\bigr{]}=\kappa_{i}\Psi^{\prime}[K_{i+j}]=\kappa_{i}\Psi_{s_{i+j}},$

which implies

$(U_{i})_{\tau_{j}}-(V_{j})_{s_{i}}+[U_{i},V_{j}]+\lambda^{j+1}(U_{i})_{\lambda}=\kappa_{i}U_{i+j},$

or, equivalently,

$U_{i}^{\prime}[\sigma_{j}]-V_{j}^{\prime}[K_{i}]+[U_{i},V_{j}]+\lambda^{j+1}(U_{i})_{\lambda}=\kappa_{i}U_{i+j},$

so that, in particular

$U_{i}^{\prime}[\sigma_{\varepsilon}]-V_{\varepsilon}^{\prime}[K_{i}]+[U_{i},V_{\varepsilon}]+\lambda^{\varepsilon+1}(U_{i})_{\lambda}=\kappa_{i}U_{i+\varepsilon},\qquad\varepsilon=-1\text{
or }0.$ (D.1)

Obviously

$\frac{\partial W_{n}}{\partial
t_{m}}=\sum_{i=1}^{n}(\mathbf{v}_{n,i})_{t_{m}}U_{i}.$

Further

$\displaystyle
W_{n}^{\prime}[\mathbb{K}_{m}]-W_{m}^{\prime}[\mathbb{K}_{n}]\overset{\eqref{Wn}}{=}V_{\varepsilon}^{\prime}[\mathbb{K}_{m}]+\sum_{i=1}^{n}\mathbf{v}_{n,i}U_{i}^{\prime}[\mathbb{K}_{m}]-V_{\varepsilon}^{\prime}[\mathbb{K}_{n}]-\sum_{j=1}^{m}\mathbf{v}_{m,j}U_{j}^{\prime}[\mathbb{K}_{n}]$
$\displaystyle\overset{\eqref{hd}}{=}\sum_{j=1}^{m}\mathbf{v}_{m,j}\left(V_{\varepsilon}^{\prime}[K_{j}]-U_{j}^{\prime}[\sigma_{\varepsilon}]\right)-\sum_{i=1}^{n}\mathbf{v}_{n,i}\left(V_{\varepsilon}^{\prime}[K_{i}]-U_{i}^{\prime}[\sigma_{\varepsilon}]\right)+\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{v}_{n,i}\mathbf{v}_{m,j}\left(U_{i}^{\prime}[K_{j}]-U_{j}^{\prime}[K_{i}]\right)$
$\displaystyle\overset{\eqref{ZCUV},\eqref{ZCU}}{=}\sum_{j=1}^{m}\mathbf{v}_{m,j}\left([U_{j},V_{\varepsilon}]+\lambda^{\varepsilon+1}(U_{j})_{\lambda}-\kappa_{j}U_{j+\varepsilon}\right)$
$\displaystyle\qquad-\sum_{i=1}^{n}\mathbf{v}_{n,i}\left([U_{i},V_{\varepsilon}]+\lambda^{\varepsilon+1}(U_{i})_{\lambda}-\kappa_{i}U_{i+\varepsilon}\right)-\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{v}_{n,i}\mathbf{v}_{m,j}[U_{i},U_{j}],$

and

$[W_{n},W_{m}]\overset{\eqref{Wn}}{=}\sum_{j=1}^{m}\mathbf{v}_{m,j}[V_{\varepsilon},U_{j}]+\sum_{i=1}^{n}\mathbf{v}_{n,i}[U_{i},V_{\varepsilon}]+\sum_{i=1}^{n}\sum_{j=1}^{m}\mathbf{v}_{n,i}\mathbf{v}_{m,j}[U_{i},U_{j}],$

and finally

$\lambda^{\varepsilon+1}(W_{n})_{\lambda}-\lambda^{\varepsilon+1}(W_{m})_{\lambda}\overset{\eqref{Wn}}{=}\lambda^{\varepsilon+1}\sum_{i=1}^{n}\mathbf{v}_{n,i}(U_{i})_{\lambda}-\lambda^{\varepsilon+1}\sum_{j=1}^{m}\mathbf{v}_{m,j}(U_{j})_{\lambda}.$

Thus, for $m<n$

$\displaystyle\frac{\partial W_{n}}{\partial
t_{m}}+W_{n}^{\prime}[\mathbb{K}_{m}]-W_{m}^{\prime}[\mathbb{K}_{n}]+[W_{n},W_{m}]+\lambda^{\varepsilon+1}(W_{n})_{\lambda}-\lambda^{\varepsilon+1}(W_{m})_{\lambda}=$
$\displaystyle\qquad=\sum_{i=1}^{n}(\mathbf{v}_{n,i})_{t_{m}}U_{i}-\sum_{j=1-\varepsilon}^{m}\kappa_{i}\mathbf{v}_{m,i}U_{i+\varepsilon}+\sum_{i=1-\varepsilon}^{n}\kappa_{i}\mathbf{v}_{n,i}U_{i+\varepsilon}$
$\displaystyle\qquad=-\varepsilon(\mathbf{v}_{n,n})_{t_{m}}U_{n}+\sum_{i=m+1}^{n}\left[(\mathbf{v}_{n,i+\varepsilon})_{t_{m}}+\kappa_{i}\mathbf{v}_{n,i}\right]U_{i+\varepsilon}$
$\displaystyle\qquad\quad+\sum_{i=1-\varepsilon}^{m}\left[(\mathbf{v}_{n,i+\varepsilon})_{t_{m}}+\kappa_{i}(\mathbf{v}_{n,i}-\mathbf{v}_{m,i})\right]U_{i+\varepsilon}=0.$

Thus, (4.17) holds if and only if

$\begin{cases}\varepsilon(\mathbf{v}_{n,n})_{t_{m}}=0,\\\
(\mathbf{v}_{n,i})_{t_{m}}+\kappa_{i-\varepsilon}\mathbf{v}_{n,i-\varepsilon}=0,&m+1+\varepsilon\leqslant
i\leqslant n+\varepsilon\\\
(\mathbf{v}_{n,i})_{t_{m}}+\kappa_{i-\varepsilon}\left(\mathbf{v}_{n,i-\varepsilon}-\mathbf{v}_{m,i-\varepsilon}\right)=0,&1\leqslant
i\leqslant m+\varepsilon\end{cases}$

which coincides with equations (3.8) for $\varepsilon=-1$ and with (3.18) for
$\varepsilon=0$. Thus, in both cases, $\mathbf{v}_{n,i}\equiv\mathbf{u}_{n,i}$
for all $n,i$.

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