Open Communications in Nonlinear Mathematical Physics ]ocnmp[ Vol.2 (2022) pp
1–References Letter

††footnotetext:  © The author(s). Distributed under a Creative Commons
Attribution 4.0 International License

Letter to the Editors

On fully-nonlinear symmetry-integrable equations with rational functions in
their highest derivative: Recursion operators

Marianna Euler and Norbert Euler${}^{\,*}$

Centro Internacional de Ciencias, Av. Universidad s/n, Colonia Chamilpa,
62210 Cuernavaca, Morelos, Mexico
∗ Corresponding author<EMAIL_ADDRESS>
Abstract: We report a class of symmetry-intergable third-order evolution
equations in 1+1 dimensions under the condition that the equations admit a
second-order recursion operator that contains an adjoint symmetry (integrating
factor) of order six. The recursion operators are given explicitly.

##  1 Introduction

We recently reported four fully-nonlinear Möbius-invariant and symmetry-
integrable third-order evolution equations, namely [2]

$\displaystyle u_{t}=\frac{u_{x}}{(b-S)^{2}},\quad b\neq 0$ (1.1a)
$\displaystyle u_{t}=\frac{u_{x}}{S^{2}}$ (1.1b) $\displaystyle
u_{t}=-2\frac{u_{x}}{\sqrt{S}}$ (1.1c) $\displaystyle
u_{t}=\frac{u_{x}(a_{1}-S)}{(a_{1}^{2}+3a_{2})(S^{2}-2a_{1}S-3a_{2})^{1/2}},\quad
a_{1}^{2}+3a_{2}\neq 0,$ (1.1d)

where $S$ denotes the Schwarzian Derivative

$\displaystyle
S:=\frac{u_{xxx}}{u_{x}}-\frac{3}{2}\left(\frac{u_{xx}}{u_{x}}\right)^{2}.$
(1.2)

This classification was achieved by matching quasi-linear auxiliary symmetry-
integrable evolution equations in $S$ for each equation (1.1a) – (1.1d). In
[3] we propose a method to compute the higher members of the hierarchies of
(1.1a) – (1.1d) without the knowledge of the equations’ recursion operators.
In particular, the proposed method makes use of the recursion operators of the
auxiliary quasi-linear evolution equations in the variable $S$. This is an
essential point since it is in general rather complicated and tedious to
compute recursion operators, especially for fully-nonlinear equations. It is
important to point out that the method to compute the higher-order members of
the hierarchies as proposed in [3], only applies to evolution equations that
are Möbius-invariant and symmetry-integrable. Furthermore we point out that it
is not possible to extend the idea of Möbius-invariant evolution equation to
systems of evolution equations in a direct sense. This has been investigated
in [4].

Inspired by the above mentioned results, we address here the problem of
identifying fully-nonlinear symmetry-integrable evolution equations beyond the
Möbius-invariant class and we do so by requiring the equations to admit a
recursion operator of a certain form. In particular, we restrict ourselves to
evolution equations that contain rational functions in $u_{xxx}$. Moreover, we
assume a recursion operator of order two with an integrating factor of maximum
order six. This of course restricts us to a special class of equations, namely
equations that admit those type of recursion operators. Nevertheless, we
believe that our findings are of interest and that the results reported here
are new.

We would like to point out that Hernández Heredero [6] classified a type of
third-order integrable fully-nonlinear evolution equations that does not
include equations with rational functions in $u_{xxx}$.

##  2 Notations and conditions

To fix the notation and to recall the conditions that are needed in this
paper, we consider the general $n$th-order autonomous evolution equation in
1+1 dimensions

$\displaystyle E:=u_{t}-F(u,u_{x},u_{xx},u_{xxx},\ldots,u_{nx})=0.$ (2.1)

The subscripts of $u$ denote partial derivatives, where partial derivatives of
order 4 and higher are indicated by $u_{nx}$, $n\geq 4$.

Equation (2.1) is said to be symmetry-integrable if it admits a recursion
operator $R[u]$ that generates an infinite number of local Lie-Bäcklund (or
generalized) symmetries for the equation. In this paper we consider recursion
operators of the following form

$\displaystyle
R[u]:=\sum_{k=1}^{m}G_{k}[u]D_{x}^{k}+G_{0}[u]+\sum_{j=1}^{p}I_{j}[u]D_{x}^{-1}\circ\Lambda_{j}[u].$
(2.2)

The notation $R[u]$ and $G_{j}[u]$ indicates that the operator $R$ and
functions $G_{j}$ depend on $u,\ u_{x},\ u_{xx},\ldots$ up to an order that is
ab initio not fixed. Here $I_{j}$ are Lie-Bäcklund symmetry coefficients for
(2.1), i.e. the coefficients of a symmetry generator

$\displaystyle Z=I_{j}[u]\frac{\partial\ }{\partial u}$ (2.3)

which satisfies the condition

$\displaystyle\left.\vphantom{\frac{DA}{DB}}L_{E}[u]I_{j}[u]\right|_{E=0}=0,$
(2.4)

where $L_{E}[u]$ denoted the linear operator

$\displaystyle L_{E}[u]:=\frac{\partial E}{\partial u}+\frac{\partial
E}{\partial u_{t}}D_{t}+\frac{\partial E}{\partial u_{x}}D_{x}+\frac{\partial
E}{\partial u_{xx}}D_{x}^{2}+\cdots+\frac{\partial E}{\partial
u_{nx}}D_{x}^{n}.$ (2.5)

$\Lambda_{j}[u]$ are integrating factors for conservation laws

$\displaystyle\left.\vphantom{\frac{DA}{DB}}D_{t}\Phi^{t}[u]+D_{x}\Phi^{x}[u]\right|_{E=0}=0,$
(2.6)

of (2.1), where

$\displaystyle\Lambda[u]=\hat{E}[u]\Phi^{t}[u]$ (2.7)

and $\Lambda$ must satisfy the condition

$\displaystyle\left.\vphantom{\frac{DA}{DB}}\hat{E}[u]\left(\Lambda[u]E\right)\right|_{E=0}=0.$
(2.8)

Here $\hat{E}[u]$ is the Euler operator

$\displaystyle\hat{E}[u]:=\frac{\partial\ }{\partial
u}-D_{t}\circ\frac{\partial\ }{\partial u_{t}}-D_{x}\circ\frac{\partial\
}{\partial u_{x}}+D_{x}^{2}\circ\frac{\partial\ }{\partial
u_{xx}}-D_{x}^{3}\circ\frac{\partial\ }{\partial u_{3x}}+\cdots.$ (2.9)

Note that condition (2.8) is equivalent to

$\displaystyle\left.\vphantom{\frac{DA}{DB}}L^{*}_{E}[u]\Lambda[u]\right|_{E=0}=0$
(2.10a) $\displaystyle\mbox{and}\quad L_{\Lambda}[u]E=L_{\Lambda}^{*}[u]E.$
(2.10b)

The first condition (2.10a) requires $\Lambda$ to be an adjoint symmetry for
(2.1), whereas the second condition (2.10b) requires $\Lambda$ to be a self-
adjoint function (for scalar evolution equations this means even-order). Here
$L_{E}^{*}[u]$ denotes the adjoint operator of $L_{E}[u]$, namely

$\displaystyle L_{E}^{*}[u]:=\frac{\partial E}{\partial
u}-D_{t}\circ\frac{\partial E}{\partial u_{t}}-D_{x}\circ\frac{\partial
E}{\partial u_{x}}+D_{x}^{2}\circ\frac{\partial E}{\partial
u_{xx}}-D_{x}^{3}\circ\frac{\partial E}{\partial u_{3x}}+\cdots.$ (2.11)

The condition on the recursion operator $R[u]$ for (2.1) is

$\displaystyle[L_{F}[u],\,R[u]]\varphi=(D_{t}R[u])\varphi,$ (2.12)

where $[\ ,\ ]$ denotes the commutator (or Lie bracket). Condition (2.12) is
evaluated on the equation (2.1). Moreover, the recursion operator of (2.1)
should generate a hierarchy of symmetries coefficients $\eta$ for (2.1), i.e.
symmetry generators of the form

$\displaystyle Z=\eta[u]\frac{\partial\ }{\partial u},$ (2.13)

by acting $R[u]$ repeatedly on $\eta$. That is

$\displaystyle R^{k}[u]\eta_{1}[u]=\eta_{k+1}[u],\qquad k=1,2,\ldots.$ (2.14)

For a symmetry-integrable evolution equation we require that all symmetries
coefficients $\eta$ generated by $R$ are local, so a recursion operator for
that equation would generate a hierarchy of local evolution equations

$\displaystyle u_{t_{k}}=R^{k}[u]F[u],\qquad k=1,2,\ldots.$ (2.15)

Each evolution equation in the hierarchy (2.15) should share the same set of
symmetries that are generated by acting the recursion operator on the first
(or seed) member for the hierarchy of (2.1). Those symmetries then span an
Abelian Lie algebra and the recursion operator is hereditary for each member
of the hierarchy (see [5] and [7] for more details).

##  3 Recursion operators for a class of third-order symmetry-integrable
equations

Our starting point is the general class of third-order autonomous evolution
equations of the form

$\displaystyle E:=u_{t}-F(u_{x},u_{xx},u_{xxx})=0.$ (3.1)

For the symmetry-integrability of (3.1) we need to establish a recursion
operator for the equation. In this paper we consider second-order recursion
operators $R[u]$ of the form

$\displaystyle
R[u]=G_{2}[u]D_{x}^{2}+G_{1}[u]D_{x}+G_{0}[u]+I_{1}[u]D_{x}^{-1}\circ\Lambda_{1}[u]+I_{2}[u]D_{x}^{-1}\circ\Lambda_{2}[u].$
(3.2)

The explicit conditions on $G_{j}$, $I_{j}$, $\Lambda_{j}$ and $F$ for (3.1)
are given in Appendix A.

In order to find equations of the form (3.1) that may admit a recursion
operator of the form (3.2), we first establish the most general form of $F$ in
terms of it highest derivative $u_{xxx}$. This is achieved by solving the
first three equations in the split commutator condition (2.12), namely those
conditions on $G_{j}$, and $F$ that do not involve the conditions on the
integrating factors $\Lambda_{j}$ or the symmetries $I_{j}$. These are the
conditions (A.2a), (A.2b) and (A.2c) given in Appendix A.

######  Proposition 1.

In terms of the variable $u_{xxx}$, the most general form of
$F(u_{x},u_{xx},u_{xxx})$ for which (3.1) admits a recursion operator of the
form (3.2) is given by the following four cases:

$\displaystyle
F=\frac{Q_{3}(u_{x},u_{xx})\left[\vphantom{\frac{DA}{DB}}u_{xxx}+Q_{2}(u_{x},u_{xx})\right]}{Q_{1}(u_{x},u_{xx})\left[\vphantom{\frac{DA}{DB}}Q_{1}(u_{x},u_{xx})+(u_{xxx}+Q_{2}(u_{x},u_{xx}))^{2}\right]^{1/2}}+Q_{4}(u_{x},u_{xx})$
(3.3a) $\displaystyle F=Q_{1}(u_{x},u_{xx})\,u_{xxx}+Q_{2}(u_{x},u_{xx})$
(3.3b) $\displaystyle
F=\frac{Q_{1}(u_{x},u_{xx})}{\left[\vphantom{\frac{DA}{DB}}u_{xxx}+Q_{2}(u_{x},u_{xx})\right]^{2}}+Q_{3}(u_{x},u_{xx})$
(3.3c) $\displaystyle
F=\frac{Q_{1}(u_{x},u_{xx})(u_{xxx}+Q_{2}(u_{x},u_{xx}))}{Q_{2}^{2}(u_{x},u_{xx})\left[\vphantom{\frac{DA}{DB}}u_{xxx}^{2}+2Q_{2}(u_{x},u_{xx})u_{xxx}\right]^{1/2}}+Q_{3}(u_{x},u_{xx}).$
(3.3d)

The functions $Q_{1},\ Q_{2},\ Q_{3}$ and $Q_{4}$ are arbitrary in their
indicated arguments.

Proof: Solving (A.2a), (A.2b) and (A.2c) we obtain the following condition on
$F(u_{x},u_{xx},$ $u_{xxx})$:

$\displaystyle
F^{\prime}\left[9(F^{\prime})^{2}F^{(4)}-45F^{\prime}F^{\prime\prime}F^{\prime\prime\prime}+40(F^{\prime\prime})^{3}\right]=0,$
(3.4)

where the primes denote partial derivatives with repect to $u_{xxx}$ and
$F^{(4)}$ the fourth partial derivative with repect to $u_{xxx}$. The general
solution of (3.4) is given by (3.3a), whereby (3.3b), (3.3c) and (3.3d) are
singular solutions. $\Box$

######  Remark 1.

We remark that the conditions given in Proposition 1 are consistent with the
conditions (2.3), (2.4) and (2.5) reported in [6].

The functions $Q_{1},\ Q_{2},\ Q_{3}$ and $Q_{4}$ should now be determined to
gain recursion operators of the form (2.2) for the equation (3.1) for each
case $F$ listed in Proposition 1. This identifies the exact form of $F$ for
the symmetry-integrability of (3.1), which is achieved by solving the
remaining conditions (A.2d), (A.2e), (A.2f) and (A.2g) given in Appendix A.

In the current paper we restrict ourselves to the case where
$F(u_{x},u_{xx},u_{xxx})$ is a rational functions in $u_{xxx}$, namely case
(3.3c). This leads to the following

######  Proposition 2.

The following equations, in the class $u_{t}=F(u_{x},u_{xx},u_{xxx})$ with $F$
a rational function in $u_{xxx}$, are symmetry-integrable:

  * •

Case I

$\displaystyle u_{t}=\frac{u_{xx}^{6}}{(\alpha
u_{x}+\beta)^{3}u_{xxx}^{2}}+Q(u_{x}),$ (3.5a) where $\\{\alpha,\ \beta\\}$
are arbitrary constants, not simultaneously zero, and $Q(u_{x})$ needs to
satisfy $\displaystyle(\alpha
u_{x}+\beta)\frac{d^{5}Q}{du_{x}^{5}}+5\alpha\frac{d^{4}Q}{du_{x}^{4}}=0,$
(3.5b) which admits for $\alpha\neq 0$ the general solution $\displaystyle
Q(u_{x})=c_{5}\left(u_{x}+\frac{\beta}{\alpha}\right)^{3}+c_{4}\left(u_{x}+\frac{\beta}{\alpha}\right)^{2}+c_{3}\left(u_{x}+\frac{\beta}{\alpha}\right)$
$\displaystyle\qquad+c_{2}\left(u_{x}+\frac{\beta}{\alpha}\right)^{-1}+c_{1}.$
(3.5c) For $\alpha=0$, the general solution of (3.5b) is $\displaystyle
Q(u_{x})=c_{5}u_{x}^{4}+c_{4}u_{x}^{3}+c_{3}u_{x}^{2}+c_{2}u_{x}+c_{1}.$
(3.5d)

Here $c_{j}$ are constants of integration.

  * •

Case II

$\displaystyle
u_{t}=\frac{u_{xx}^{3}\left(\lambda_{1}+\lambda_{2}u_{xx}\right)^{3}}{u_{xxx}^{2}},$
(3.6)

where $\\{\lambda_{1},\ \lambda_{2}\\}$ are arbitrary constants but not
simultaneously zero.

  * •

Case III

$\displaystyle u_{t}=\frac{(\alpha
u_{x}+\beta)^{11}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{2}},$ (3.7)

where $\\{\alpha,\ \beta\\}$ are arbitrary constants but not simultaneously
zero.

  * •

Case IV

$\displaystyle
u_{t}=\frac{4u_{x}^{5}}{(2b\,u_{x}^{2}-2u_{x}u_{xxx}+3u_{xx}^{2})^{2}}\equiv\frac{u_{x}}{(b-S)^{2}},$
(3.8)

where $b$ is an arbitrary constant and $S$ is the Schwarzian derivative (1.2).

The recursion operators for each equation listed in Proposition 2 have been
computed and are given in Appendix B. Note that equation (3.8) is identical to
the Möbius-invariant equation (1.1a). This recursion operator for equation
(1.1b) is obtained by setting $b=0$ in the recursion operator (B.7) of (3.8).

For each equation listed in Proposition 2 one can easily remove the
nonlinearity in the third derivative by a simple substitution $u_{x}=W(x,t)$
which, in a sense, “unpotentialises” the equations of Proposition 2. For
completeness, we list the so obtained equations here:

  * •

Case I: With $u_{x}=W(x,t)$, (3.5a) takes the form

$\displaystyle W_{t}=-\frac{2W_{x}^{6}W_{xxx}}{(\alpha
W+\beta)^{3}W_{xx}^{3}}-\frac{3\alpha W_{x}^{7}}{(\alpha
W+\beta)^{4}W_{xx}^{2}}+\frac{6W_{x}^{5}}{(\alpha W+\beta)^{3}W_{xx}}$
$\displaystyle\qquad+Q^{\prime}(W)W_{x},$ (3.9a) where $\displaystyle(\alpha
W+\beta)Q^{(5)}+5\alpha Q^{(4)}=0,\quad Q=Q(W).$ (3.9b)

  * •

Case II: With $u_{x}=W_{1}(x,t)$, we obtain for (3.6) the following equation:

$\displaystyle
W_{1,t}=-\frac{2W_{1,x}^{3}(\lambda_{1}+\lambda_{2}W_{1,x})^{3}W_{1,xxx}}{W_{1,xx}^{3}}+\frac{3\lambda_{2}W_{1,x}^{3}(\lambda_{1}+\lambda_{2}W_{1,x})^{2}}{W_{1,xx}}$
$\displaystyle\qquad+\frac{3W_{1,x}^{2}(\lambda_{1}+\lambda_{2}W_{1,x})^{3}}{W_{1,xx}}.$
(3.10)

With $W_{1,x}=W_{2}(x,t)$, we obtain for (• ‣ 3) the following equation:

$\displaystyle
W_{2,t}=-\frac{2W_{2}^{2}(\lambda_{1}+\lambda_{2}W_{2})^{3}W_{2,xxx}}{W_{2,x}^{3}}+\frac{6W_{2}^{3}(\lambda_{1}+\lambda_{2}W_{2})^{3}W_{2,xx}^{2}}{W_{2,x}^{4}}$
$\displaystyle\qquad-\frac{9W_{2}^{2}(\lambda_{1}+\lambda_{2}W_{2})^{3}(W_{2}+1)W_{2,xx}}{W_{2,x}^{2}}+\frac{9\lambda_{2}W_{2}^{2}(\lambda_{1}+\lambda_{2}W_{2})^{2}(W_{2,x}+1)}{W_{2,x}}$
$\displaystyle\qquad-6\lambda_{2}^{2}W_{2}^{3}(\lambda_{1}+\lambda_{2}W_{2})+6W_{2}(\lambda_{1}+\lambda_{2}W_{2})^{3}.$
(3.11)

  * •

Case III: With $u_{x}=W(x,t)$, we obtain for (3.7) the following equation:

$\displaystyle W_{t}=-\frac{(\alpha
W+\beta)^{10}}{\left[\vphantom{\frac{DA}{DB}}(\alpha v+\beta)W_{xx}-3\alpha
W_{x}^{2}\right]^{3}}\left[\vphantom{\frac{DA}{DB}}2\alpha
WW_{xxx}\left(\alpha W+2\beta\right)+2\beta^{2}W_{xxx}\right.$
$\displaystyle\qquad\qquad\left.\vphantom{\frac{DA}{B}}-21\alpha
W_{x}W_{xx}\left(\alpha W+\beta\right)+33\alpha^{2}W_{x}^{3}\right].$ (3.12)

  * •

Case IV: With $u_{x}=W(x,t)$, we obtain for (3.8) the following equation:

$\displaystyle
W_{t}=\frac{4W^{4}}{\left(\vphantom{\frac{DA}{DB}}2bW^{2}-2WW_{xx}+3W_{x}^{2}\right)^{3}}\left(\vphantom{\frac{DA}{DB}}4W^{2}W_{xxx}-18WW_{x}W_{xx}\right.$
$\displaystyle\qquad\qquad\left.\vphantom{\frac{DA}{DB}}+15W_{x}^{3}+2bW^{2}W_{x}\right).$
(3.13)

##  4 Concluding remarks

Our aim has been to construct fully-nonlinear third-order evolution equations
in the class $u_{t}=F(u_{x},u_{xx},u_{xxx})$, namely to identify those
equations in this class that admit a second-order recursion operator with a
sixth-order integrating factor, which are then symmetry-integrable equations.
Note that that exists no fully-nonlinear evolution equation in this class that
admits a recursion operator of order two where both integrating factors,
$\Lambda_{1}$ and $\Lambda_{2}$, are of order less than six.

We report here four equations, listed in Proposition 2, namely (3.5a), (3.6),
(3.7) and (3.8). Due to the mentioned restrictions on the form of the
recursion operator, this is certainly not a complete classification of all
fully-nonlinear third-order evolution equations of this form that admit a
recursion operator. Nevertheless, we do consider the equations that we have
obtained here to be of some interest and worthy of further study. It would,
for example, be interesting to find all the potentialisations of the four
fully-nonlinear equations (3.5a) to (3.8), as well as the equations (3.9a) to
(3.13). This can be investigated by using the adjoint symmetries structure of
the equations. Some preliminary calculations have revealed a rich adjoint
symmetry structure for these equations, so one can expect to obtain
interesting results. Furthermore, one could apply the multi-potentialisation
method which may lead to nonlocal symmetries for the equations (see [1] for
details regarding multi-potentialisations). One could also extend this study
further, namely to include evolution equations of third order that explicitly
depend on $u$ and allow algebraic functions in $u_{xxx}$.

### Acknowledgements

We thank the anonymous referee for useful remarks regarding the classification
of integrable evolution equations, for point out some misprints, and for
clarifying the results in [6].

##  Appendix A Appendix: The general conditions for $R[u]$ of (3.1)

For the equation $u_{t}=F(u_{x},u_{xx},u_{xxx})$ we provide here the explicit
general conditions on the functions $F$, $G_{j}$, $I_{j}$ and $\Lambda_{j}$
for the existence of a recursion operator $R[u]$ of the form

$\displaystyle
R[u]=G_{2}[u]D_{x}^{2}+G_{1}[u]D_{x}+G_{0}[u]+I_{1}[u]D_{x}^{-1}\circ\Lambda_{1}[u]+I_{2}[u]D_{x}^{-1}\circ\Lambda_{2}[u].$
(A.1)

This is obtained from the commutator condition (2.12) by equating to zero all
the derivatives of the free function $\varphi$. For convenience we introduce
the following notation:

$\displaystyle A_{1}:=\frac{\partial F}{\partial u_{x}},\qquad
A_{2}:=\frac{\partial F}{\partial u_{xx}},\qquad A_{3}:=\frac{\partial
F}{\partial u_{xxx}}.$

The conditions are as follows:

$\displaystyle\frac{\partial^{4}\varphi}{\partial
x^{4}}:-2G_{2}D_{x}A_{3}+3A_{3}D_{x}G_{2}=0$ (A.2a)
$\displaystyle\frac{\partial^{3}\varphi}{\partial
x^{3}}:2A_{2}D_{x}G_{2}-2G_{2}D_{x}A_{2}-G_{2}D_{x}^{2}A_{3}-G_{1}D_{x}A_{3}+3A_{3}D_{x}^{2}G_{2}$
$\displaystyle\qquad\ +3A_{3}D_{x}G_{1}=0$ (A.2b)
$\displaystyle\frac{\partial^{2}\varphi}{\partial
x^{2}}:3A_{3}D_{x}^{2}G_{1}+A_{3}D_{x}^{3}G_{2}+2A_{2}D_{x}G_{1}+A_{2}D_{x}^{2}G_{2}+A_{1}D_{x}G_{2}+3A_{3}D_{x}G_{0}$
$\displaystyle\qquad\
\left.\vphantom{\frac{DA}{DB}}-2G_{2}D_{x}A_{1}-G_{2}D_{x}^{2}A_{2}-G_{1}D_{x}A_{2}-D_{t}G_{2}\right|_{E=0}=0$
(A.2c) $\displaystyle\frac{\partial\varphi}{\partial
x}:A_{3}D_{x}^{3}G_{1}+3A_{3}D_{x}^{2}G_{0}+A_{2}D_{x}^{2}G_{1}+2A_{2}D_{x}G_{0}+A_{1}D_{x}G_{1}$
$\displaystyle\qquad\
+\sum_{j=1}^{2}\left(\vphantom{\frac{DA}{DB}}3A_{3}\Lambda_{j}D_{x}I_{j}+3A_{3}I_{j}D_{x}\Lambda_{j}+I_{j}\Lambda_{j}D_{x}A_{3}\right)$
$\displaystyle\qquad\
\left.\vphantom{\frac{DA}{DB}}-G_{2}D_{x}^{2}A_{1}-G_{1}D_{x}A_{1}-D_{t}G_{1}\right|_{E=0}=0$
(A.2d)
$\displaystyle\varphi:A_{3}D_{x}^{3}G_{0}+A_{2}D_{x}^{2}G_{0}+A_{1}D_{x}G_{0}+\sum_{j=1}^{2}\left(\vphantom{\frac{DA}{DB}}-2I_{j}(D_{x}\Lambda_{j})(D_{x}A_{3})-I_{j}\Lambda_{j}D_{x}^{2}A_{3}\right.$
$\displaystyle\qquad\
\left.+I_{j}\Lambda_{j}D_{x}A_{2}-I_{j}D_{x}^{4}\Lambda_{j}+3A_{3}(D_{x}I_{j})(D_{x}\Lambda_{j})+3A_{3}\Lambda_{j}D_{x}^{2}I_{j}\right.$
$\displaystyle\qquad\
\left.\left.\vphantom{\frac{DA}{DB}}+2A_{2}\Lambda_{j}D_{x}I_{j}+2A_{2}I_{j}D_{x}\Lambda_{j}\right)-D_{t}G_{0}\right|_{E=0}=0,$
(A.2e) as well as the symmetry condition
$\displaystyle\left.\vphantom{\frac{DA}{DB}}L_{E}[u]\,I_{j}\right|_{E=0}=0,\qquad
j=1,2$ (A.2f) and the adjoint symmetry condition
$\displaystyle\left.\vphantom{\frac{DA}{DB}}L^{*}_{E}[u]\,\Lambda_{j}\right|_{E=0}=0,\qquad
j=1,2.$ (A.2g)

##  Appendix B Appendix: The recursion operators for the symmetry-integrable
equations of Proposition 2

Recursion operator for Case I: Equation (3.5a) of Proposition 2 viz.

$\displaystyle u_{t}=\frac{u_{xx}^{6}}{(\alpha
u_{x}+\beta)^{3}u_{xxx}^{2}}+Q(u_{x}),$

admits the recursion operator

$\displaystyle R[u]=G_{2}[u]D_{x}^{2}+G_{1}[u]D_{x}+G_{0}[u]+(\alpha
u_{x}+\beta)D_{x}^{-1}\circ\Lambda_{1}[u],$ (B.1)

where

$\displaystyle G_{2}[u]=\frac{u_{xx}}{(\alpha u_{x}+\beta)^{2}\,u_{xxx}^{2}}$
(B.2a) $\displaystyle G_{1}[u]=\frac{u_{xx}^{4}u_{4x}}{(\alpha
u_{x}+\beta)^{2}\,u_{xxx}^{3}}-\frac{4u_{xx}^{3}}{(\alpha
u_{x}+\beta)^{2}\,u_{xxx}}+\frac{\alpha u_{xx}^{5}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{2}}$ (B.2b) $\displaystyle
G_{0}[u]=-\frac{u_{xx}^{4}u_{5x}}{(\alpha
u_{x}+\beta)^{2}\,u_{xxx}^{3}}+\frac{3u_{xx}^{4}u_{4x}^{2}}{(\alpha
u_{x}+\beta)^{2}\,u_{xxx}^{4}}$
$\displaystyle\qquad+\left(-\frac{8u_{xx}^{3}}{(\alpha
u_{x}+\beta)^{2}u_{xxx}^{2}}+\frac{6\alpha u_{xx}^{5}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{3}}\right)u_{4x}+\frac{6\alpha^{2}u_{xx}^{6}}{(\alpha
u_{x}+\beta)^{4}\,u_{xxx}^{2}}$ $\displaystyle\qquad-\frac{18\alpha
u_{xx}^{4}}{(\alpha u_{x}+\beta)^{3}\,u_{xxx}}+\frac{12\alpha
u_{xx}^{2}}{(\alpha u_{x}+\beta)^{2}}+\frac{1}{3}(\alpha
u_{x}+\beta)\frac{d^{2}Q}{du_{x}^{2}}-\frac{\alpha}{3}\frac{dQ}{du_{x}}$
(B.2c) $\displaystyle\Lambda_{1}=\frac{u_{xx}^{4}u_{6x}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{3}}+\left(\frac{12u_{xx}^{3}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{2}}-\frac{9\alpha u_{xx}^{5}}{(\alpha
u_{x}+\beta)^{4}\,u_{xxx}^{3}}\right)u_{5x}$
$\displaystyle\qquad+\left(\frac{24\alpha u_{xx}^{2}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}}-\frac{72\alpha u_{xx}^{4}}{(\alpha
u_{x}+\beta)^{4}\,u_{xxx}^{2}}+\frac{36\alpha^{2}u_{xx}^{6}}{(\alpha
u_{x}+\beta)^{5}\,u_{xxx}^{3}}\right)u_{4x}$
$\displaystyle\qquad-\frac{9u_{xx}^{4}u_{4x}u_{5x}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{4}}+\left(\frac{27\alpha u_{xx}^{5}}{(\alpha
u_{x}+\beta)^{4}\,u_{xxx}^{4}}-\frac{28u_{xx}^{3}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{3}}\right)u_{4x}^{2}$
$\displaystyle\qquad+\frac{12u_{xx}^{4}u_{4x}^{3}}{(\alpha
u_{x}+\beta)^{3}\,u_{xxx}^{5}}-\frac{24u_{xx}u_{xxx}}{(\alpha
u_{x}+\beta)^{3}}+\frac{30\alpha^{3}u_{xx}^{7}}{(\alpha
u_{x}+\beta)^{6}\,u_{xxx}^{2}}+\frac{108\alpha u_{xx}^{3}}{(\alpha
u_{x}+\beta)^{4}}$ $\displaystyle\qquad-\frac{108\alpha^{2}u_{xx}^{5}}{(\alpha
u_{x}+\beta)^{5}\,u_{xxx}}-\frac{1}{2}\frac{d^{3}Q}{du_{x}^{3}}\,u_{xx}.$
(B.2d)

Here $Q(u_{x})$ needs to satisfy the 5th-order ordinary differential equation
(3.5b), viz.

$\displaystyle(\alpha
u_{x}+\beta)\frac{d^{5}Q}{du_{x}^{5}}+5\alpha\frac{d^{4}Q}{du_{x}^{4}}=0.$

Recursion operator for Case II: Equation (3.6) of Proposition 2 viz.

$\displaystyle
u_{t}=\frac{u_{xx}^{3}\left(\lambda_{1}+\lambda_{2}u_{xx}\right)^{3}}{u_{xxx}^{2}}$

admits the recursion operator

$\displaystyle
R[u]=G_{2}[u]D_{x}^{2}+G_{1}[u]D_{x}+G_{0}[u]+D_{x}^{-1}\circ\Lambda_{1}[u],$
(B.3)

where

$\displaystyle
G_{2}[u]=\frac{u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}}{u_{xxx}^{2}}$
(B.4a) $\displaystyle
G_{1}[u]=\frac{u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}u_{4x}}{u_{xxx}^{3}}-\frac{4\lambda_{2}u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})}{u_{xxx}}$
(B.4b) $\displaystyle
G_{0}[u]=\frac{u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}u_{5x}}{u_{xxx}^{3}}+\frac{3u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}u_{4x}^{2}}{u_{xxx}^{4}}$
$\displaystyle\qquad-\frac{2u_{xx}u_{xxx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})(\lambda_{1}+4\lambda_{2}u_{xx})u_{4x}}{u_{xxx}^{4}}+12\lambda_{2}^{2}u_{xx}^{2}+6\lambda_{1}\lambda_{2}u_{xx}$
(B.4c)
$\displaystyle\Lambda_{1}[u]=\frac{u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}u_{6x}}{u_{xxx}^{3}}+\frac{4u_{xx}(\lambda_{1}+\lambda_{2}u_{xx})(\lambda_{1}+3\lambda_{2}u_{xx})u_{5x}}{u_{xxx}^{2}}$
$\displaystyle\qquad-\frac{9u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}u_{4x}u_{5x}}{u_{xxx}^{4}}+\frac{12u_{xx}^{2}(\lambda_{1}+\lambda_{2}u_{xx})^{2}u_{4x}^{3}}{u_{xxx}^{5}}$
$\displaystyle\qquad-\frac{2u_{xx}u_{xxx}^{3}(\lambda_{1}+\lambda_{2}u_{xx})(5\lambda_{1}+14\lambda_{2}u_{xx})u_{4x}^{2}}{u_{xxx}^{5}}$
$\displaystyle\qquad+\frac{2(12\lambda_{2}^{2}u_{xx}^{2}+10\lambda_{1}\lambda_{2}u_{xx}+\lambda_{1}^{2})u_{4x}}{u_{xxx}}-6\lambda_{2}(\lambda_{1}+4\lambda_{2}u_{xx})u_{xxx}.$
(B.4d)

Recursion operator for Case III: Equation (3.7) of Proposition 2 viz.

$\displaystyle u_{t}=\frac{(\alpha
u_{x}+\beta)^{11}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{2}}$

admits the recursion operator

$\displaystyle R[u]=G_{2}[u]D_{x}^{2}+G_{1}[u]D_{x}+G_{0}[u]+(\alpha
u_{x}+\beta)D_{x}^{-1}\circ\Lambda_{1}[u]$ (B.5)

where

$\displaystyle G_{2}[u]=\frac{(\alpha
u_{x}+\beta)^{8}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{2}}$ (B.6a) $\displaystyle
G_{1}[u]=\frac{(\alpha
u_{x}+\beta)^{7}\,u_{4x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha
u_{xx}^{2}\right]^{3}}\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)^{2}u_{4x}-13\alpha(\alpha u_{x}+\beta)u_{xx}u_{xxx}\right.$
$\displaystyle\qquad\qquad\left.\vphantom{\frac{DA}{DB}}+24\alpha^{2}u_{xx}^{3}\right]$
(B.6b) $\displaystyle G_{0}[u]=-\frac{(\alpha
u_{x}+\beta)^{9}u_{5x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{3}}$
$\displaystyle\qquad+\frac{3(\alpha
u_{x}+\beta)^{6}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha
u_{xx}^{2}\right]^{4}}\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)^{4}u_{4x}^{2}-\frac{46}{3}\alpha(\alpha
u_{x}+\beta)^{3}u_{xx}u_{xxx}u_{4x}\right.$ $\displaystyle\qquad\qquad\
+3\alpha(\alpha u_{x}+\beta)^{3}u_{xxx}^{3}+\frac{184}{3}\alpha^{2}(\alpha
u_{x}+\beta)^{2}u_{xx}^{2}u_{xxx}^{2}$
$\displaystyle\qquad\qquad\left.\vphantom{\frac{DA}{DB}}-184\alpha^{3}(\alpha
u_{x}+\beta)u_{xx}^{4}u_{xxx}+144\alpha^{4}u_{xx}^{6}\right]$ (B.6c)
$\displaystyle\Lambda_{1}[u]=\frac{(\alpha
u_{x}+\beta)^{8}\,u_{6x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{3}}-\frac{9(\alpha
u_{x}+\beta)^{9}\,u_{4x}u_{5x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{4}}$
$\displaystyle\qquad-\frac{72\alpha^{2}(\alpha
u_{x}+\beta)^{7}\,u_{xx}^{3}u_{5x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{4}}+\frac{81\alpha(\alpha
u_{x}+\beta)^{8}\,u_{xx}u_{xxx}u_{5x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{4}}$
$\displaystyle\qquad+\frac{12(\alpha
u_{x}+\beta)^{10}\,u_{4x}^{3}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{5}}$
$\displaystyle\qquad-\frac{45\alpha(\alpha
u_{x}+\beta)^{8}\,u_{xx}u_{4x}^{2}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha u_{xx}^{2}\right]^{5}}\left(5\alpha
u_{x}u_{xxx}+5\beta u_{xxx}-3\alpha u_{xx}^{2}\right)$
$\displaystyle\qquad+\frac{5\alpha(\alpha
u_{x}+\beta)^{6}\,u_{4x}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha
u_{xx}^{2}\right]^{5}}\left[\vphantom{\frac{DA}{DB}}11(\alpha
u_{x}+\beta)^{3}u_{xxx}^{3}+291\alpha(\alpha
u_{x}+\beta)^{2}u_{xx}^{2}u_{xxx}^{2}\right.$
$\displaystyle\qquad\qquad\left.\vphantom{\frac{DA}{DB}}-504\alpha^{2}(\alpha
u_{x}+\beta)u_{xx}^{4}u_{xxx}+216\alpha^{3}u_{xx}^{6}\right]$
$\displaystyle\qquad-\frac{20\alpha^{2}(\alpha
u_{x}+\beta)^{4}\,u_{xx}}{\left[\vphantom{\frac{DA}{DB}}(\alpha
u_{x}+\beta)u_{xxx}-3\alpha
u_{xx}^{2}\right]^{7}}\left[\vphantom{\frac{DA}{DB}}-\frac{67\alpha^{2}}{3}(\alpha
u_{x}+\beta)^{4}u_{xx}^{4}u_{xxx}^{4}\right.$
$\displaystyle\qquad\qquad+148\alpha^{3}(\alpha
u_{x}+\beta)^{3}u_{xx}^{6}u_{xxx}^{3}+288\alpha^{5}(\alpha
u_{x}+\beta)u_{xx}^{10}u_{xxx}$
$\displaystyle\qquad\qquad-306\alpha^{4}(\alpha
u_{x}+\beta)^{2}u_{xx}^{8}u_{xxx}^{2}+\frac{31}{27}(\alpha
u_{x}+\beta)^{6}u_{xxx}^{6}$ $\displaystyle\qquad\quad\
\,\left.\vphantom{\frac{DA}{DB}}-\frac{38\alpha}{9}(\alpha
u_{x}+\beta)^{5}u_{xx}^{2}u_{xxx}^{5}-108\alpha^{6}u_{xx}^{12}\right].$ (B.6d)

Recursion operator for Case VI: Equation (3.8) of Proposition 2 viz.

$\displaystyle
u_{t}=\frac{4u_{x}^{5}}{(2b\,u_{x}^{2}-2u_{x}u_{xxx}+3u_{xx}^{2})^{2}}\equiv\frac{u_{x}}{(b-S)^{2}},$

admits the recursion operator

$\displaystyle
R[u]=G_{2}[u]D_{x}^{2}+G_{1}[u]D_{x}+G_{0}[u]+u_{x}D_{x}^{-1}\circ\Lambda_{1}[u]+u_{t}D_{x}^{-1}\circ\Lambda_{2}[u]$
(B.7)

where

$\displaystyle G_{2}[u]=\frac{1}{4(b-S)^{2}}$ (B.8a) $\displaystyle
G_{1}[u]=-\frac{u_{xx}}{2u_{x}(b-S)^{2}}-\frac{S_{x}}{4(b-S)^{3}}$ (B.8b)
$\displaystyle
G_{0}[u]=\frac{u_{xx}^{2}}{8u_{x}^{2}(b-S)^{2}}+\frac{u_{xx}S_{x}}{4u_{x}(b-S)^{3}}+\frac{S_{xx}}{4(b-S)^{3}}-\frac{2bS^{2}-b^{2}S-3S_{x}^{2}-S^{3}}{4(b-S)^{4}}$
(B.8c)
$\displaystyle\Lambda_{1}[u]=-\frac{S_{xxx}}{4u_{x}(b-S)^{3}}-\frac{9S_{x}S_{xx}}{4u_{x}(b-S)^{4}}-\frac{S_{x}(b+3S)}{8u_{x}(b-S)^{3}}-\frac{3S_{x}^{3}}{u_{x}(b-S)^{5}}$
(B.8d) $\displaystyle\Lambda_{2}[u]=-\frac{S_{x}}{8u_{x}}.$ (B.8e)

Here $S$ is the Schwarzian derivative (1.2).

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