#  $\mathcal{C}^{\infty}$-symmetries of distributions and integrability
††thanks: Citation: Authors. Title. Pages…. DOI:000000/11111.

*A.J. Pan-Collantes, A. Ruiz, C. Muriel, J.L. Romero
Departamento de Matemáticas
Universidad de Cádiz - UCA
Puerto Real
<EMAIL_ADDRESS>adrian.ruiz@uca.es,concepcion.muriel<EMAIL_ADDRESS>

###### Abstract

An extension of the notion of solvable structure for involutive distributions
of vector fields is introduced. The new structures are based on a
generalization of the concept of symmetry of a distribution of vector fields,
inspired in the extension of Lie point symmetries to
$\mathcal{C}^{\infty}$-symmetries for ODEs developed in the recent years.
These new objects, named $\mathcal{C}^{\infty}$-structures, play a fundamental
role in the integrability of the distribution: the knowledge of a
$\mathcal{C}^{\infty}$-structure for a corank $k$ involutive distribution
permits to find its integral manifolds by solving $k$ successive completely
integrable Pfaffian equations. These results have important consequences for
the integrability of differential equations. In particular, we derive a new
procedure to integrate an $m$th-order ordinary differential equation by
splitting the problem into $m$ completely integrable Pfaffian equations. This
step-by-step integration procedure is applied to integrate completely several
equations that cannot be solved by standard procedures.

_K_ eywords symmetry of a distribution $\cdot$ solvable structure $\cdot$
$\mathcal{C}^{\infty}$-symmetry of a distribution $\cdot$
$\mathcal{C}^{\infty}$-structure $\cdot$ Frobenius integrability $\cdot$
differential equations

##  1 Introduction

Note: The published version of this preprint is in Journal of Differential
Equations Volume 348, 5 March 2023, Pages 126-153.

It is well known that for a first-order ordinary differential equation (ODE)
Lie symmetry method permits integration by quadrature when a symmetry is known
[1, 2, 3, 4]. Indeed, the knowledge of a Lie point symmetry for a first-order
ODE is equivalent to finding a first integral and its corresponding
integrating factor. For $m$th-order equations, a one-parameter group of Lie
point symmetries permits to reduce the order by one; once the reduced equation
has been solved, the solution of the original equation can be constructed by a
quadrature. Similarly, a $r$-parameter group of Lie point symmetries with a
solvable Lie algebra reduces to a $(m-r)$th-order ODE, and the general
solution of the equation can be recovered by $r$ quadratures from the solution
of the reduced equation. When $m=r$, the equation can completely be solved
after $m$ quadratures.

However, the existence of a solvable group of Lie point symmetries is not a
necessary condition for the integrability by quadratures. Indeed, there exist
many examples of ODEs with a trivial group of Lie point symmetries that can be
integrated [2, 5, 6, 7]. This suggests that there must exist objects beyond
Lie point symmetries that explain such integrability processes. Among then,
solvable structures and $\mathcal{C}^{\infty}$-symmetries have played an
important role in problems of integrability of ODEs in the recent literature.

In 1991 Basarab-Horwath characterized the integrability by quadratures by
using the concepts of symmetry and solvable structure [8, 9] for involutive
distributions of vector fields. When applied to the integration of a given
ODE, the procedure requires starting with a symmetry $X$ of the distribution
spanned by the vector field $A$ associated to the ODE, i.e., $X$ must satisfy
the condition

$[X,A]=\rho A$ (1)

for some smooth function $\rho.$ The procedure pursues with a symmetry of the
distribution generated by $\\{A,X\\}$ and continues by using an additional
symmetry of the subsequent distributions in each stage. In this context, the
vector fields $X$ satisfying (1) correspond to generalized symmetries [2] of
the given ODE, also called dynamical symmetries [3] or higher-order symmetries
[1].

On the other hand, the concept of $\mathcal{C}^{\infty}$-symmetry (or
$\lambda$-symmetry) was introduced in [10], as a generalization of the notion
of Lie point symmetry. That concept is based on a new method of prolonging a
vector field by using a smooth function $\lambda.$ The resulting
$\lambda$-prolonged vector field $X$ satisfies a weaker condition than in (1):

$[X,A]=\lambda X+\rho A.$ (2)

Only when $\lambda=0,$ $X$ is a symmetry of the distribution generated by $A$.
Like Lie point symmetries, $\mathcal{C}^{\infty}$-symmetries can be used to
reduce the order or to find first integrals of ODEs. In this way there are
more chances to achieve order reductions or exact solutions, even in the
absence of Lie point symmetries [2, 5, 6, 7].

Inspired by these two approaches, in Section 3 we firstly introduce the
concept of $\mathcal{C}^{\infty}$-symmetry of an involutive distribution of
vector fields $\mathcal{Z}$. Secondly, the concept of solvable structure is
extended by using $\mathcal{C}^{\infty}$-symmetries of distributions instead
of symmetries. In this way we achieve a generalization of the notion of
solvable structure, which will be called a structure of
$\mathcal{C}^{\infty}$-symmetries (or simply a
$\mathcal{C}^{\infty}$-structure) for $\mathcal{Z}$. We also provide the
concept of $\mathcal{C}^{\infty}$-structure of 1-forms for $\mathcal{Z},$
establishing the correspondence between these new two concepts (Proposition
3.1). After that, we investigate how the main result in [8], concerning the
integrability by quadratures through solvable structures, can be generalized
when $\mathcal{C}^{\infty}$-structures are considered. It is proven that the
knowledge of a $\mathcal{C}^{\infty}$-structure for a corank $k$ involutive
distribution permits to find the integral manifolds by solving $k$ completely
integrable Pfaffian equations (Theorem 3.1).

Since the elements of a $\mathcal{C}^{\infty}$-structure satisfy weaker
conditions than those required for a solvable structure, the new approach
enlarges the class of vector fields that can be used to integrate involutive
distributions. On the other hand, it must be said that a drawback is that the
Pfaffian equations associated to a $\mathcal{C}^{\infty}$-structure, although
completely integrable, might not be defined by closed 1-forms, as in the case
of solvable structures.

These results are firstly established in the general context of distributions
and later applied in Section 4 to integrability problems of differential
equations. In particular, we prove that for a $m$th-order ODE, the knowledge
of a $\mathcal{C}^{\infty}$-structure permits to split the ODE into $m$
completely integrable Pfaffian equations.

The new approach, apart from its theoretical interest, broadens the class of
vector fields that can be involved in the process of integrating differential
equations, enlarging the range of problems that can be addressed. Indeed, as
we will show in Section 4, a $\mathcal{C}^{\infty}$-structure can be used to
integrate ODEs that present some obstacle when are attempted to be solved by
using standard procedures.

The examples in this section aim to show how $\mathcal{C}^{\infty}$-structures
can be constructed and applied to solve this type of problems. The equation
studied in Example 4.1 admits just one Lie point symmetry. The Lie symmetry
method fails because the quadrature that recovers the solutions from the
reduced equation (which is of Abel type) cannot be explicitly evaluated. For
the equations in examples 4.2 and 4.3 the Lie symmetry method cannot be
applied because the equations lack Lie point symmetries. All of them can be
completely integrated by using the $\mathcal{C}^{\infty}$-structure approach.
Finally, we consider in Example 4.4 a non-autonomous first-order system of two
ordinary differential equations to illustrate how the method can also be used
for systems of ODEs.

##  2 Preliminaries

In this section we introduce some notation and theoretical background of our
work.

###  2.1 Basic assumptions and notations

Due to the local character of our results, we will work in an open simply
connected subset $U$ of an euclidean space $\mathbb{R}^{n}$. All the results
can be extended straightforwardly to the case of an $n$-dimensional manifold.
The vector fields, differential forms and functions are assumed to be smooth
(meaning $\mathcal{C}^{\infty}$) although this condition could be relaxed for
each particular situation. We also assume that the domains have been
restricted when necessary.

By $\mathfrak{X}(U)$ and ${\Omega}^{k}(U)$ we will denote the modules over
$\mathcal{C}^{\infty}(U)$ of all smooth vector fields and $k$-forms,
respectively, while $\Omega^{*}(U)$ will denote the exterior algebra of all
the differential forms on $U$ [11, 12].

Given $r$ pointwise linearly independent vector fields
$Z_{1},\ldots,Z_{r}\in\mathfrak{X}(U),$ we will denote by

$\mathcal{Z}:=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ (3)

the submodule of $\mathfrak{X}(U)$ generated by $Z_{1},\ldots,Z_{r}.$
Similarly, the submodule of $\Omega^{1}(U)$ generated by a set of $r$
pointwise linearly independent 1-forms
$\Lambda=\\{\omega_{1},\ldots,\omega_{r}\\}$ will be denoted by
$\mathcal{S}(\Lambda).$

The submodule $\mathcal{Z}$ (resp. $\mathcal{S}(\Lambda)$) defines a
distribution (resp. a Pfaffian differential system) of constant rank $r$.

A $k$-form $\omega$ is said to annihilate $\mathcal{Z}$ if
$\omega(Y_{1},\ldots,Y_{k})=0$ on $U$ whenever
$Y_{1},\dots,Y_{k}\in\mathcal{Z}$ [12, Definition 2.26]; we denote

$\mbox{Ann}(\mathcal{Z})=\\{\omega\in\Omega^{*}(U):\omega\quad\mbox{annihilates}\quad\mathcal{Z}\\}.$
(4)

The set $\mbox{Ann}(\mathcal{Z})$ is an ideal in $\Omega^{*}(U)$ which is
locally generated by $n-r$ pointwise linearly independent 1-forms
$\Lambda:=\\{\omega_{1},\ldots,\omega_{n-r}\\}$ [12, Proposition 2.28]. In
this case we will write

$\mbox{Ann}(\mathcal{Z})=\mathcal{I}(\Lambda).$ (5)

The associated submodule $\mathcal{S}(\Lambda)$ will be denoted by
$\mathcal{Z}^{\circ}$ and it can be easily checked that

$\mathcal{Z}^{\circ}=\mathcal{S}(\Lambda)=\\{\omega\in\Omega^{1}(U):Z\,\lrcorner\,\omega=0,\,\mbox{for
each}\,Z\in\mathcal{Z}\\},$ (6)

where $\left.\right\lrcorner$ denotes contraction (interior product). By
linearity, it is clear that $\omega\in\mathcal{Z}^{\circ}$ (resp.
$Z\in\mathcal{Z}$) if and only if $Z_{k}\,\lrcorner\,\omega=0$ for $1\leq
k\leq r$ (resp. $Z\,\lrcorner\,\omega_{l}=0$ for $1\leq l\leq n-r$).

The distribution $\mathcal{Z}$ is said to be involutive if it is closed under
the Lie bracket. This involutivity can be characterized in terms of the
algebraic ideal (5). For the following proposition, let us recall that an
ideal $\mathcal{J}\subset\Omega^{*}(U)$ is called a differential ideal [12,
Definition 2.29] if it is is closed under exterior differentiation $d;$ that
is $d(\mathcal{J})\subset\mathcal{J}.$

######  Proposition 2.1.

[12, Proposition 2.30] A distribution $\mathcal{Z}$ is involutive if and only
if the ideal $\mbox{Ann}(\mathcal{Z})=\mathcal{I}(\Lambda)$ is a differential
ideal.

A smooth function $F$ on $U$ is a first integral of $\mathcal{Z}$ if $Z(F)=0$
for each $Z\in\mathcal{Z}$ or, equivalently, if $Z_{i}(F)=0$ for $1\leq i\leq
r.$ Similarly, we say that $F$ is a first integral of $\mathcal{S}(\Lambda)$
if $dF\in\mathcal{S}(\Lambda)$ or, equivalently, if

$dF\wedge\omega_{1}\wedge\ldots\wedge\omega_{n-r}=0.$ (7)

Therefore, a function $F$ is a first integral of $\mathcal{Z}$ if and only if
$F$ is a first integral of $\mathcal{Z}^{\circ}.$

###  2.2 Distributions and solvable structures

For an involutive distribution (3), Frobenius Theorem establishes the local
existence of a complete set of first integrals of $\mathcal{Z},$ i.e., $n-r$
functionally independent first integrals of $\mathcal{Z}.$ The integral
manifolds111In this paper integral manifolds of distributions will always be
integral manifolds of maximal dimension [12, Remark 1.58]. of $\mathcal{Z}$
can be implicitly defined by the corresponding level sets. In fact, Frobenius
Theorem guarantees, for $p\in U,$ the existence of a unique maximal [12,
Definition 1.63] connected integral manifold of the involutive distribution
through $p$ [12, Theorem 1.64]. However, Frobenius Theorem does not provide a
procedure to find the integral manifolds.

An important notion concerning the search for integral manifolds of
distributions is that of solvable structure, which is based on the following
concept of symmetry of a distribution [8, 9, 13, 14, 15, 16, 17]:

######  Definition 2.1.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution of vector fields on an open set $U\subset\mathbb{R}^{n}.$ A
vector field $X$ on $U$ is called a symmetry of $\mathcal{Z}$ if :

  1. 1.

$\\{Z_{1},\ldots,Z_{r},X\\}$ is a set of pointwise linearly independent vector
fields on $U;$ and

  2. 2.

there exist smooth functions $c_{ik}$ on $U$ such that

$[X,Z_{i}]=\sum_{k=1}^{r}c_{ik}Z_{k}\text{, }\text{ for }i=1,\ldots,r.$

The previous notion of symmetry has been used in [8] to define a solvable
structure as follows:

######  Definition 2.2.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution on an open subset $U$ of $\mathbb{R}^{n}$. Let $\langle
X_{1},\ldots,X_{n-r}\rangle$ be an ordered collection of vector fields on $U$
and denote

$\begin{array}[]{l}\mathcal{X}_{0}:=\\{Z_{1},\ldots,Z_{r}\\}\quad\mbox{and}\quad\mathcal{X}_{k}:=\\{Z_{1},\ldots,Z_{r},X_{1},\ldots,X_{k}\\},\quad\mbox{for}\quad
1\leq k\leq n-r.\end{array}$ (8)

We will say that $\langle X_{1},\ldots,X_{n-r}\rangle$ is a solvable structure
for $\mathcal{Z}$ if for $1\leq k\leq n-r$ the vector field $X_{k}$ is a
symmetry of the rank $r+k-1$ distribution $\mathcal{S}(\mathcal{X}_{k-1}).$

######  Remark 2.1.

  1. (a)

In references such as [8, 9, 13, 14, 15, 16] the term solvable structure
refers to the whole ordered set $\langle
Z_{1},\ldots,Z_{r},X_{1},\ldots,X_{n-r}\rangle,$ while in this work the term
solvable structure for $\mathcal{Z}$ refers only to the part $\langle
X_{1},\ldots,X_{n-r}\rangle,$ because the distribution $\mathcal{Z}$ is fixed
and given in advance.

  2. (b)

Observe that in Definition 2.2, for $0\leq k\leq n-r,$ the vector fields in
$\mathcal{X}_{k}$ are pointwise linearly independent on $U.$ In particular,
$X_{1},\ldots,X_{n-r}$ are also pointwise linearly independent.

The knowledge of a solvable structure provides a systematic procedure to find
the integral manifolds of the distribution $\mathcal{Z}$ by integrating
successively certain 1-forms constructed from the elements of the solvable
structure. At each stage there appears a 1-form that can be integrated by
quadrature. This is the main result on solvable structures concerning the
integrability of the distribution (see [8, 9, 13, 14, 15, 16] for further
details):

######  Proposition 2.2.

[8, Proposition 3] Given a solvable structure for an involutive distribution
$\mathcal{Z}$, their integral manifolds can be found, at least locally, by
quadratures alone.

######  Remark 2.2.

Here the term quadrature refers to the problem of finding a smooth function
$F$ such that locally $dF=\omega$, for a previously given exact 1-form
$\omega$. Such function $F$ is called a primitive of $\omega.$ Since (7) is
satisfied, $F$ is a first integral of $\mathcal{S}(\\{\omega\\}).$ The level
sets of $F$ are integral manifolds of the distribution annihilated by
$\mathcal{S}(\\{\omega\\}).$

For further reference, let us recall that if $\omega$ is not exact but at
least $\omega\wedge d\omega=0$ then $\mathcal{I}(\\{\omega\\})$ is a
differential ideal and the distribution associated to the Pfaffian system
$\mathcal{S}(\\{\omega\\})$ is involutive. By Frobenius Theorem there also
exist (at least locally) integral manifolds described by the level sets of a
smooth function $F$. Therefore there also exists a non-vanishing smooth
function $\mu$, such that $dF=\mu\omega$. In this case it is said that
$\omega\equiv 0$ is a _completely integrable Pfaffian equation_ or that the
1-form $\omega$ is _Frobenius integrable_. The function $\mu$ is called
integrating factor of $\omega$ and $F$ is a first integral of
$\mathcal{S}(\\{\omega\\})$.

###  2.3 Symmetries and solvable structures for ODEs

In this section we adapt the former concepts and results to the framework of
the integrability of a given $m$th-order ODE

$u_{m}=\phi(x,u,u_{1},\ldots,u_{m-1}),$ (9)

where $(x,u)$ are in some open set $M\subset\mathbb{R}^{2}$ and represent the
independent and dependent variables, respectively, and $u_{i}$ denotes the
derivative of $u$ with respect to $x$ of order $i$, for $i=1,\dots,m.$ The
corresponding $m$th-order jet space [18] is denoted by
$J^{m}(\mathbb{R},\mathbb{R})$ and $\phi$ is a smooth function defined on the
open subset $M^{(m-1)}\subseteq J^{m-1}(\mathbb{R},\mathbb{R}).$

Equation (9) determines the rank 1 distribution $\mathcal{S}(\\{A\\})$ defined
by the vector field

$A=\partial_{x}+u_{1}\partial_{u}+\ldots+\phi\partial_{u_{m-1}}\in\mathfrak{X}(M^{(m-1)}).$
(10)

The distribution $\mathcal{S}(\\{A\\})$ is trivially involutive and hence
integrable. Its integral manifolds (curves) correspond to solutions of (9)
prolonged to the $(m-1)$th-order jet space.

The Definition 2.1 of symmetry of a distribution, when applied to
$\mathcal{S}(\\{A\\})$, includes well-known notions of symmetry of an ODE.
Given a Lie point symmetry of (9)

$\mathbf{v}=\xi(x,u)\partial_{x}+\eta(x,u)\partial_{u},$ (11)

let $X:=\mathbf{v}^{(m-1)}$ denote the standard $(m-1)$th-order prolongation
of $\mathbf{v}$ [2]. The vector field $X$ satisfies the relation [3]:

$[X,A]=\rho A,$ (12)

where $\rho=-A(\xi).$ Therefore, according to Definition 2.1, $X$ is a
symmetry of the distribution $\mathcal{S}(\\{A\\}).$ When the functions $\xi$
and $\eta$ in (11) also depend on derivatives of $u$ we obtain _generalized_
symmetries [2] (also called _dynamical_ symmetries [3] or _higher-order_
symmetries [1]). They satisfy the same condition (12) [3, Chapter 12] and are
therefore symmetries in the sense of distributions.

In this context, a solvable structure for $\mathcal{S}(\\{A\\})$ consists of
an ordered collection of vector fields $\langle X_{1},\ldots,X_{m}\rangle$
such that:

  1. (i)

$X_{1}$ is a generalized symmetry of (9).

  2. (ii)

For $2\leq k\leq m$ the vector field $X_{k}$ is a symmetry, in the sense of
Definition 2.1, of the distribution
$\mathcal{S}(\\{A,X_{1},\ldots,X_{k-1}\\})$.

It must be observed that in general the vector fields $X_{k}$ for $2\leq k\leq
m$ are not necessarily generalized symmetries of the equation. On the other
hand, the $m$ vector fields of a solvable structure do not necessarily
constitute a solvable Lie algebra. However, a $m$-dimensional solvable algebra
of symmetries provides a particular case of solvable structure [8, 9, 13, 14,
15].

As stated in Proposition 2.2, the knowledge of a solvable structure is
sufficient to integrate the equation by quadratures. The reader is referred to
[8, 9, 13, 14, 15, 16] for more details on the procedure.

##  3 $\mathcal{C}^{\infty}$-structure for a distribution

Although most of the classical procedures to integrate a given ODE are
consequence of the existence of Lie point symmetries, there are many ODEs
without Lie point symmetries that can be integrated. This motivated the search
of a wider class of vector fields that could explain such integration
procedures. In this context, $\mathcal{C}^{\infty}$-symmetries (or
$\lambda$-symmetries) [10] and its posterior generalizations [19, 20, 21, 22,
23, 24, 25, 26] emerged as extensions of the Lie point symmetry concept. The
notion of $\mathcal{C}^{\infty}$-symmetry is based on a different way of
prolonging a given vector field (11), which uses a smooth function $\lambda.$
If a pair $(\mathbf{v},\lambda)$ is a $\mathcal{C}^{\infty}$-symmetry of the
equation (9) then the $(m-1)$th-order $\lambda$-prolongation of $\mathbf{v}$
denoted by $X:=\mathbf{v}^{[\lambda,(m-1)]}$ [10], satisfies

$[X,A]=\lambda X+\rho A,$ (13)

which provides a weaker condition than condition (12).

When instead of vector fields of the form (11), generalized vector fields are
considered, the concept of $\mathcal{C}^{\infty}$-symmetry can be extended in
the following straightforward way:

######  Definition 3.1.

A vector field $X\in\mathfrak{X}(M^{(m-1)})$ will be called a generalized
$\mathcal{C}^{\infty}$-symmetry of equation (9) if there exist
$\lambda,\rho\in\mathcal{C}^{\infty}(M^{(m-1)})$ such that

$[X,A]=\lambda X+\rho A.$

Our next goal is to use a similar idea in order to introduce the notion of
$\mathcal{C}^{\infty}$-symmetry of a distribution.

######  Definition 3.2.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution on $U.$ A vector field $X$ on $U$ is called a
$\mathcal{C}^{\infty}$-symmetry of $\mathcal{Z}$ if:

  1. 1.

$\\{Z_{1},\ldots,Z_{r},X\\}$ is a set of pointwise linearly independent vector
fields on $U;$ and

  2. 2.

the distribution $\mathcal{S}(\\{Z_{1},\ldots,Z_{r},X\\})$ is involutive,
i.e., there exist smooth functions $\lambda_{i},c_{ik}$ on $U$
($i,k=1,\ldots,r$) such that

$[X,Z_{i}]=\lambda_{i}X+\sum_{k=1}^{r}c_{ik}Z_{k}\text{, }i=1,\ldots,r.$ (14)

######  Remark 3.1.

  1. (a)

The notion of symmetry of a distribution in Definition 2.1 is a particular
case of the concept given in Definition 3.2. It corresponds to the particular
case when $\lambda_{i}=0$ for $i=1,\ldots,r.$

  2. (b)

The notion of $\lambda$-symmetry of a distribution in [26, Definition 2] is a
particular case of the concept given in Definition 3.2. It corresponds to the
particular case when $\lambda_{i}=\lambda_{j}$ for $i,j=1,\ldots,r.$

  3. (c)

A $\mathcal{C}^{\infty}$-symmetry of the distribution
$\mathcal{Z}=\mathcal{S}(\\{A\\})$ generated by the vector field (10)
associated to equation (9) is a generalized $\mathcal{C}^{\infty}$-symmetry of
(9) in the sense of Definition 3.1.

  4. (d)

If $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{n-1}\\})$ is an involutive
distribution of corank 1 then $X$ is a $\mathcal{C}^{\infty}$-symmetry of
$\mathcal{Z}$ if and only if $\\{Z_{1},\ldots,Z_{n-1},X\\}$ is a set of
pointwise linearly independent vector fields.

The following concept extends the notion of solvable structure (Definition
2.2) when we use $\mathcal{C}^{\infty}$-symmetries instead of symmetries:

######  Definition 3.3.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution on an open subset $U$ of $\mathbb{R}^{n}$. Let $\langle
X_{1},\ldots,X_{n-r}\rangle$ be an ordered collection of vector fields on $U$
and define the sets $\mathcal{X}_{k},$ for $0\leq k\leq n-r,$ as in (8). We
will say that $\langle X_{1},\ldots,X_{n-r}\rangle$ is a structure of
$\mathcal{C}^{\infty}$-symmetries for $\mathcal{Z}$, or simply a
$\mathcal{C}^{\infty}$-structure for $\mathcal{Z}$, if for $1\leq k\leq n-r$
the vector field $X_{k}$ is a $\mathcal{C}^{\infty}$-symmetry of the rank
$r+k-1$ distribution $\mathcal{S}(\mathcal{X}_{k-1}).$

######  Remark 3.2.

  1. (a)

Condition 1 in Definition 3.2 implies that for $0\leq k\leq n-r$ the sets
$\mathcal{X}_{k}$ appearing in Definition 3.3 (and in particular
$\\{X_{1},\ldots,X_{n-r}\\}$) are sets of pointwise linearly independent
vector fields on $U.$

  2. (b)

In Definition 3.3 the distributions $\mathcal{S}(\mathcal{X}_{k})$ are
necessarily involutive for $k=0,\ldots,n-r.$ In fact, $\langle
X_{1},\ldots,X_{n-r}\rangle$ is a $\mathcal{C}^{\infty}$-structure for
$\mathcal{Z}$ if and only if for $1\leq k\leq n-r$ the distribution
$\mathcal{S}(\mathcal{X}_{k})$ is involutive and $\mathcal{X}_{n-r}$ is a set
of pointwise linearly independent vector fields on $U.$

  3. (c)

A solvable structure is a particular case of a
$\mathcal{C}^{\infty}$-structure, in which all the vector fields $X_{i}$ are
symmetries of the corresponding distribution.

  4. (d)

The last element of a $\mathcal{C}^{\infty}$-structure can be any vector field
$X_{n-r}$ such that $\mathcal{X}_{n-r}$ is a set of pointwise linearly
independent vector fields on $U$ (see Remark 3.1 (d)).

By using Proposition 2.1, the conditions on the involutivity of
$\mathcal{S}(\mathcal{X}_{k}),$ for $0\leq k\leq n-r,$ (see Remark 3.2 (b)) in
Definition 3.3 can also be expressed in terms of the associated Pfaffian
system $\mathcal{S}(\mathcal{X}_{k})^{\circ}.$ This motivates the following
definition:

######  Definition 3.4.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution on an open set $U\subset\mathbb{R}^{n}$. Let
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ be an ordered collection of
pointwise linearly independent 1-forms on $U$ and define the sets

$\Lambda_{k}:=\\{\omega_{k+1},\ldots,\omega_{n-r}\\}\quad\mbox{for}\quad 0\leq
k\leq n-r-1.$ (15)

We will say that $\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ is a
$\mathcal{C}^{\infty}$-structure of 1-forms for $\mathcal{Z}$ if the following
two conditions hold:

  1. (a)

$\mathcal{Z}^{\circ}=\mathcal{S}\left(\Lambda_{0}\right)$.

  2. (b)

$\mathcal{I}(\Lambda_{k})$ is a differential ideal for $k=1,\ldots,n-r-1$.

The following proposition states that there exists a correspondence between
the objects introduced in Definition 3.3 and Definition 3.4:

######  Proposition 3.1.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution on an open subset $U$ of $\mathbb{R}^{n}$. Given a
$\mathcal{C}^{\infty}$-structure $\langle X_{1},\ldots,X_{n-r}\rangle$ for
$\mathcal{Z}$, there exists a $\mathcal{C}^{\infty}$-structure of 1-forms
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ such that

$\mathcal{S}\left(\mathcal{X}_{k}\right)^{\circ}=\mathcal{S}\left(\Lambda_{k}\right),\quad\mbox{for}\quad
1\leq k\leq n-r-1$ (16)

and conversely. When conditions (16) are satisfied we will say that $\langle
X_{1},\ldots,X_{n-r}\rangle$ and
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ are $\mathcal{Z}$-related.

###### Proof.

Given a $\mathcal{C}^{\infty}$-structure $\langle X_{1},\ldots,X_{n-r}\rangle$
for $\mathcal{Z}$, consider the ordered set
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ of the last $n-r$ elements of
the dual frame of $\mathcal{X}_{n-r}$ and define the corresponding sets given
in (15). By construction, the elements in $\Lambda_{0}$ are pointwise linearly
independent and, by duality, $\mathcal{Z}^{\circ}=\mathcal{S}(\Lambda_{0})$
and conditions (16) hold. Since for $1\leq k\leq n-r-1$ the distribution
$\mathcal{S}\left(\mathcal{X}_{k}\right)$ is involutive, then
$\mathcal{I}(\Lambda_{k})$ is a differential ideal (see Proposition 2.1). This
proves that $\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ is a
$\mathcal{C}^{\infty}$-structure of 1-forms for $\mathcal{Z}$ satisfying
conditions (16).

Conversely, given a $\mathcal{C}^{\infty}$-structure of 1-forms
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ for $\mathcal{Z},$ let
$\\{\zeta_{1},\ldots,\zeta_{r}\\}$ be a set of 1-forms that completes
$\Lambda_{0}$ to a coframe
$\theta:=\\{\zeta_{1},\ldots,\zeta_{r},\omega_{1},\ldots,\omega_{n-r}\\}$ on
$U.$ Consider the ordered set $\langle X_{1},\ldots,X_{n-r}\rangle$ of the
last $n-r$ elements of the dual basis of $\theta$ and define the corresponding
sets given in (8). By construction

$X_{i}\,\lrcorner\,\omega_{j}=0\text{ on }U,\text{ for }1\leq i,j\leq
n-r,\,i\neq j.$ (17)

Since $\mathcal{Z}^{\circ}=\mathcal{S}(\Lambda_{0}),$ it follows from (17)
that conditions (16) hold.

In order to prove that $\mathcal{X}_{n-r}$ is a set of pointwise linearly
independent vector fields on $U,$ observe that if there exists $p\in U$ such
that

$\alpha_{1}({Z_{1}})_{p}+\ldots+\alpha_{r}({Z_{r}})_{p}+\alpha_{r+1}({X_{1}})_{p}+\ldots+\alpha_{n}({X_{n-r}})_{p}=0$
(18)

for some constants $\alpha_{i}\in\mathbb{R},$ then after contraction with
$({\omega_{i}})_{p},$ we obtain that $\alpha_{r+i}=0,$ for $1\leq i\leq n-r.$
Since $Z_{1},\ldots,Z_{r}$ are pointwise linearly independent vector fields on
$U,$ this implies that also $\alpha_{i}=0,$ for $1\leq i\leq r.$

Finally, the involutivity of the distributions
$\mathcal{S}\left(\mathcal{X}_{k}\right)$ follows from Proposition 2.1. ∎

######  Remark 3.3.

  1. (a)

Observe that the elements of two $\mathcal{Z}$-related
$\mathcal{C}^{\infty}$-structures do not necessarily satisfy
$X_{i}\,\lrcorner\,\omega_{j}=\delta_{ij},$ for $1\leq i,j\leq n-r.$
Conditions (16) only imply that $X_{i}\,\lrcorner\,\omega_{j}=0$ for $1\leq
i<j\leq n-r.$ Besides, since we have that
$\omega_{i}\in\mathcal{S}(\mathcal{X}_{i-1})^{\circ}$ and
$\omega_{i}\notin\mathcal{S}(\mathcal{X}_{i})^{\circ},$ then it can be checked
that, for $1\leq i\leq n-r,$ the function $X_{i}\,\lrcorner\,\omega_{i}$ does
not vanish on $U.$

  2. (b)

If $\langle X_{1},\ldots,X_{n-r}\rangle$ and
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ are $\mathcal{Z}$-related and
$h$ is any non-vanishing function on $U,$ then it can be checked that $\langle
X_{1},\ldots,hX_{i_{0}},\ldots,X_{n-r}\rangle$ and
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ are also $\mathcal{Z}$-related.
As a consequence, since the functions
$h_{i}=1/(X_{{i}}\,\lrcorner\,\omega_{{i}})$ for $1\leq i\leq n-r$ are well
defined on $U,$ then we have that $\langle
h_{1}X_{1},\ldots,h_{n-r}X_{n-r}\rangle$ and
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ are $\mathcal{Z}$-related and
satisfy the conditions $(h_{i}X_{i})\,\lrcorner\,\omega_{i}=1$ for $1\leq
i\leq n-r.$

  3. (c)

In the context of Definition 3.4, the condition
$d(\mathcal{I}(\Lambda_{k}))\subseteq\mathcal{I}(\Lambda_{k})$ is satisfied
for $0\leq k\leq n-r.$ More explicitly, for $1\leq i\leq n-r,$ there exist
1-forms $\sigma_{i}$ such that

$d\omega_{i}=\sigma_{i}\wedge\omega_{i}+\sum_{j=i+1}^{n-r}\sigma_{ij}\wedge\omega_{j},$
(19)

for some 1-forms $\sigma_{ij},$ $j=i+1,\ldots,n-r.$

The notion of solvable structure (Definition 2.2) can also be formulated in
terms of differential forms [8, 9]. In this case the corresponding conditions
become $d(\mathcal{I}(\Lambda_{k}))\subseteq\mathcal{I}(\Lambda_{k+1})$ for
$k=0,\ldots,n-r-2,$ and $d(\mathcal{I}(\Lambda_{n-r-1}))=0,$ which are more
restrictive than in the case of $\mathcal{C}^{\infty}$-structures. In other
words, for a solvable structure, conditions (19) become

$d\omega_{i}=\sum_{j=i+1}^{n-r}\sigma_{ij}\wedge\omega_{j}.$ (20)

This discussion explains, in terms of 1-forms, why
$\mathcal{C}^{\infty}$-structures require weaker conditions than solvable
structures to be determined.

The following theorem states the main result on the application of
$\mathcal{C}^{\infty}$-structures to the integration of involutive
distributions:

######  Theorem 3.1.

Let $\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ be an involutive
distribution of vector fields on $U\subset\mathbb{R}^{n}.$ Any
$\mathcal{C}^{\infty}$-structure for $\mathcal{Z}$ can be used to find the
integral manifolds of $\mathcal{Z}$ by solving successively $n-r$ completely
integrable Pfaffian equations.

###### Proof.

By Proposition 3.1 the $\mathcal{C}^{\infty}$-structure for $\mathcal{Z}$ can
be given in terms of vector fields $\langle X_{1},\ldots,X_{n-r}\rangle$ or by
a $\mathcal{C}^{\infty}$-structure of 1-forms
$\langle\omega_{1},\ldots,\omega_{n-r}\rangle$ satisfying conditions (16)
(i.e., $\mathcal{Z}$-related).

According to condition (b) in Definition 3.4, the ideal
$\mathcal{I}(\Lambda_{n-r-1})$ is a differential ideal. Therefore
$\omega_{n-r}\wedge d\omega_{n-r}=0$ and hence the Pfaffian equation

$\omega_{n-r}\equiv 0$

is completely integrable. Given $q\in U,$ let $\Sigma_{n-r}$ be a
$(n-1)$-dimensional integral manifold of $\mathcal{S}(\\{\omega_{n-r}\\})$
passing through $q$ [12, Theorem 2.32]. Consider a local parametrization of
$\Sigma_{n-r}$ defined by a smooth imbedding [12, Definition 1.27]
$\iota:\bar{U}\to U$ on some open set $\bar{U}\subset\mathbb{R}^{n-1}.$ We
define

$\bar{\omega}_{i}:=\iota^{*}(\omega_{i})\,{\text{ for }}1\leq i\leq n-r.$ (21)

Since $\Sigma_{n-r}$ is an integral manifold of
$\mathcal{S}(\\{\omega_{n-r}\\})$ then
$\bar{\omega}_{n-r}=\iota^{*}(\omega_{n-r})=0.$ On the other hand, since
$\mathcal{S}(\\{\omega_{n-r}\\})=\mathcal{S}(\mathcal{X}_{n-r-1})^{\circ},$
the vector fields

$\begin{array}[]{ll}\bar{Z_{i}}:=(\iota^{-1})_{*}\left({Z_{i}|}_{{\Sigma}_{n-r}}\right)&\,{\text{
for }}1\leq i\leq r,\\\
\bar{X_{j}}:=(\iota^{-1})_{*}\left({X_{j}|}_{{\Sigma}_{n-r}}\right)&\,{\text{
for }}1\leq j\leq n-r-1\end{array}$ (22)

are well defined. It follows from (19) and $\bar{\omega}_{n-r}=0$ that, for
$1\leq i\leq n-r-1,$

$d\bar{\omega}_{i}=\iota^{*}(d\omega_{i})=\bar{\sigma}_{i}\wedge\bar{\omega}_{i}+\displaystyle\sum_{j=i+1}^{n-r-1}\bar{\sigma}_{ij}\wedge\bar{\omega}_{j},$

where we have written $\bar{\sigma}_{i}:=\iota^{*}(\sigma_{i}),$ and
$\bar{\sigma}_{ij}:=\iota^{*}(\sigma_{ij}).$ This proves that
$\langle\bar{\omega}_{1},\ldots,\bar{\omega}_{n-r-1}\rangle$ is a
$\mathcal{C}^{\infty}$-structure of 1-forms for
$\bar{\mathcal{Z}}=\mathcal{S}(\bar{Z}_{1},\ldots,\bar{Z}_{r})$ on
$\bar{U}\subset\mathbb{R}^{n-1}.$ It can be checked that, by construction,
$\langle\bar{X}_{1},\ldots,\bar{X}_{n-r-1}\rangle$ is a
$\mathcal{C}^{\infty}$-structure of $\bar{\mathcal{Z}}$ and that it is
$\bar{\mathcal{Z}}$-related to
$\langle\bar{\omega}_{1},\ldots,\bar{\omega}_{n-r-1}\rangle.$

In consequence, $\mathcal{I}(\\{\bar{\omega}_{n-r-1}\\})$ is a differential
ideal; hence the Pfaffian equation

$\bar{\omega}_{n-r-1}\equiv 0$

is completely integrable. As before, let $\bar{\Sigma}_{n-r-1}$ denote an
$(n-2)$-dimensional integral manifold of
$\mathcal{S}(\\{\bar{\omega}_{n-r-1}\\})$ passing through
$\bar{q}=\iota^{-1}(q).$ By construction,
${\Sigma}_{n-r-1}:=\iota(\bar{\Sigma}_{n-r-1})$ is a submanifold of
${\Sigma}_{n-r}$ and it is an integral manifold of
$\mathcal{S}(\\{\omega_{n-r-1},\omega_{n-r}\\})=\mathcal{S}(\mathcal{X}_{n-r-2})^{\circ}.$

This process can be continued by using the restriction of
$\bar{\omega}_{n-r-2}$ to $\bar{\Sigma}_{n-r-1},$ which also defines a
completely integrable Pfaffian equation in a $(n-2)$-dimensional space. The
corresponding integral manifold is of dimension $n-3$ and it can imbedded into
${\Sigma}_{n-r-1}$ (and hence into ${\Sigma}_{n-r}$) through the composition
of the local parametrizations of $\bar{\Sigma}_{n-r-1}$ and ${\Sigma}_{n-r}.$
By construction, this $(n-3)$-dimensional submanifold is an integral manifold
of $\mathcal{S}(\Lambda_{n-r-3})=\mathcal{S}(\mathcal{X}_{n-r-3})^{\circ}.$ In
the last step we obtain a $r$-dimensional integral manifold of
$\mathcal{S}(\Lambda_{0})=\mathcal{S}(\mathcal{Z})^{\circ}.$ ∎

Observe that the previous proof provides a systematic procedure to construct
the integral manifolds. We refer to the reader to Example (4.3) (for instance)
to see how this constructive procedure works in practice.

Theorem 3.1 extends the class of objects that can be used to integrate
involutive distributions because $\mathcal{C}^{\infty}$-structures satisfy
weaker conditions than solvable structures (see Remark 3.3 (c)). The cost is
that the corresponding Pfaffian equations given in Theorem 3.1, although
completely integrable, might not be defined by closed 1-forms, as in the case
of solvable structures.

##  4 $\mathcal{C}^{\infty}$-structures and differential equations

###  4.1 $\mathcal{C}^{\infty}$-structures for ODEs: General case

In this section we apply the results obtained in Section 3 to the particular
case in which the involutive distribution $\mathcal{Z}$ corresponds to the
distribution $\mathcal{S}(\\{A\\})$ generated by the vector field (10)
associated to an $m$th-order ordinary differential equation of the form (9).

A $\mathcal{C}^{\infty}$-structure $\langle X_{1},\ldots,X_{m}\rangle$ for
$\mathcal{S}(\\{A\\})$ will be simply called a
$\mathcal{C}^{\infty}$-structure of equation (9). According to Definition 3.3,
the vector field $X_{1}$ must be a $\mathcal{C}^{\infty}$-symmetry of the
distribution $\mathcal{S}(\\{A\\}),$ i.e., a generalized
$\mathcal{C}^{\infty}$-symmetry of (9) in the sense of Definition 3.1 (see
Remark 3.1 (c)):

$[X_{1},A]=\lambda^{1}X_{1}+c^{1}A,$ (23)

for some functions $\lambda^{1},c^{1}\in\mathcal{C}^{\infty}(M^{(m-1)}).$ The
second vector field $X_{2}$ in the $\mathcal{C}^{\infty}$-structure must
satisfy relationships of the form:

$\begin{array}[]{rcl}\left[X_{2},A\right]&=&\lambda^{2}X_{2}+d_{1}^{2}X_{1}+c^{2}A,\\\
\left[X_{2},X_{1}\right]&=&\gamma_{1}^{2}X_{2}+\widetilde{d}_{1}^{2}X_{1}+\widetilde{c}_{1}^{2}A.\end{array}$
(24)

Similarly, for the remaining elements in the $\mathcal{C}^{\infty}$-structure,
$X_{i},$ for $i=1,\ldots,m,$ we must have

$\begin{array}[]{rcl}[X_{i},A]&=&\lambda^{i}X_{i}+d_{1}^{i}X_{1}+\ldots+d_{i-1}^{i}X_{i-1}+c^{i}A,\\\
[X_{i},X_{j}]&=&\gamma_{j}^{i}X_{i}+\widetilde{d}_{1}^{i}X_{1}+\ldots+\widetilde{d}_{i-1}^{i}X_{i-1}+\widetilde{c}_{j}^{i}A\quad\mbox{for}\quad
j<i,\end{array}$ (25)

where the coefficients are smooth functions on $M^{(m-1)}.$

As a consequence of Remark 3.1 (d), the last vector field $X_{m}$ can be any
vector field pointwise linearly independent of $\\{A,X_{1},\ldots,X_{m-1}\\}.$

The immediate application of Theorem 3.1 yields the following theorem.

######  Theorem 4.1.

A $\mathcal{C}^{\infty}$-structure for a $m$th-order equation (9) permits to
integrate the equation by solving successively $m$ completely integrable
Pfaffian equations.

To apply this theorem we can proceed as follows. Let
$\boldsymbol{\Omega}=dx\wedge du\wedge\ldots\wedge du_{m-1}$ be the volume
form in $M^{(m-1)}.$ Given the $\mathcal{C}^{\infty}$-structure $\langle
X_{1},\ldots,X_{m}\rangle$ for $\mathcal{S}(\\{A\\}),$ we construct the
following 1-forms:

$\begin{array}[]{lll}\omega_{i}&=&{X_{1}\,\lrcorner\,X_{2}\,\lrcorner\,\ldots\,\lrcorner\,\widehat{X_{i}}\,\lrcorner\,\ldots\,\lrcorner\,X_{m}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}},\quad
1\leq i\leq m,\end{array}$ (26)

where $\widehat{X_{i}}$ indicates omission of $X_{i}.$ It can be checked that
$\langle\omega_{1},\ldots,\omega_{m}\rangle$ is a
$\mathcal{C}^{\infty}$-structure of 1-forms for $\mathcal{S}(\\{A\\})$ in the
sense of Definition 3.4, and that, for $1\leq k\leq m,$ the condition
$\mathcal{S}(\mathcal{X}_{k})^{\circ}=\mathcal{S}(\Lambda_{k})$ is satisfied.
As a consequence, the ideal $\mathcal{I}(\\{\omega_{m}\\})$ is a differential
ideal, i.e.,

$\omega_{m}\wedge d\omega_{m}=0.$

Therefore, there exists a first integral $I_{m}$ of
$\mathcal{S}(\Lambda_{m-1}),$ that according to (7) satisfies

$dI_{m}\wedge\omega_{m}=0.$ (27)

Since $A\,\lrcorner\,\omega_{m}=0,$ the function $I_{m}$ is a first integral
of $A.$ Also note that $X_{k}(I_{m})=0$ for $1\leq k\leq m-1.$ The
corresponding level sets of $I_{m}$ define integral manifolds $\Sigma_{m}$ of
the involutive distribution
$\mathcal{S}(\mathcal{X}_{m-1})=\mathcal{S}(\\{A,X_{1},\ldots,X_{m-1}\\}).$

Since $\mathcal{I}(\\{\omega_{m-1},\omega_{m}\\})$ is a differential ideal,
the restriction of the 1-form $\omega_{m-1}$ to $\Sigma_{m},$ which will be
denoted by $\bar{\omega}_{m-1},$ is Frobenius integrable. As before, a first
integral $\bar{I}_{m-1}$ of $\bar{\omega}_{m-1},$ arises from the condition

$d\bar{I}_{m}\wedge\bar{\omega}_{m-1}=0.$ (28)

The procedure continues in a similar way, by integrating the remaining
completely integrable Pfaffian equations defined by the successive
restrictions of the 1-forms in the $\mathcal{C}^{\infty}$-structure. Observe
that at each stage the completely integrable Pfaffian equation has one less
variable than in the previous step.

In particular, the integrability by quadratures in terms of solvable
structures (see Section 2.3) follows from Theorem 4.1 for the special case in
which all the vector fields $X_{i}$ in the $\mathcal{C}^{\infty}$-structure
are symmetries of $\mathcal{S}(\mathcal{X}_{i-1}),$ for $1\leq i\leq m.$

###  4.2 $\mathcal{C}^{\infty}$-structures for second-order ODEs

For a second-order ODE

$u_{2}=\phi(x,u,u_{1}),$ (29)

the first element $X_{1}$ of a $\mathcal{C}^{\infty}$-structure can be given
by the first-order $\lambda$-prolongation of a known
$\mathcal{C}^{\infty}$-symmetry of the equation. By considering its canonical
representative $(\partial_{u},\lambda),$ it can be assumed without loss of
generality that
$X_{1}=(\partial_{u})^{[\lambda,(1)]}=\partial_{u}+\lambda\partial{u_{1}}$ and
hence $[X_{1},A]=\lambda X_{1}$ (see [27] for details). According to Theorem
4.1, the corresponding 1-form $\omega_{2}$ defined in (26),

$\omega_{2}=X_{1}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}=(u_{1}\lambda-\phi)dx-\lambda
du+du_{1},$ (30)

is Frobenius integrable. It can be checked that the condition
$\omega_{2}\wedge d\omega_{2}=0$ implies that function
$\lambda=\lambda(x,u,u_{1})$ must be a solution of

$\lambda_{x}+\lambda_{u}u_{1}+\lambda_{u_{1}}\phi+\lambda^{2}=\phi_{u}+\lambda\phi_{u_{1}}$
(31)

and conversely. Condition (31) is precisely the determining equation that must
satisfy $\lambda$ in order to $(\partial_{u},\lambda)$ be a
$\mathcal{C}^{\infty}$-symmetry of equation (29) [27].

By Remark 3.2 (d) the second element of a $\mathcal{C}^{\infty}$-structure can
be any vector field $X_{2}$ such that $\\{A,X_{1},X_{2}\\}$ is pointwise
linearly independent. For instance, $X_{2}=\partial_{u_{1}}.$ In this case,
$X_{2}\,\lrcorner\,X_{1}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}=\boldsymbol{\Omega}(A,X_{1},X_{2})=1.$
For the $\mathcal{C}^{\infty}$-structure $\langle X_{1},X_{2}\rangle$ the
corresponding 1-form (26) becomes

$\begin{array}[]{ll}\omega_{1}&={X_{2}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}}=u_{1}dx-
du.\end{array}$ (32)

In consequence, $\langle\omega_{1},\omega_{2}\rangle$ is a
$\mathcal{C}^{\infty}$-structure of 1-forms for equation (29), where the
1-forms $\omega_{1}$ and $\omega_{2}$ are given in (32) and (30),
respectively.

In examples 4.1 and 4.2 we use this $\mathcal{C}^{\infty}$-structure to
integrate second-order ODEs by using Theorem 4.1.

The equation in Example 4.1 admits just one Lie point symmetry, that can be
used to reduce the second-order equation to an Abel-type equation. For most of
this kind of problems, the classical Lie method fails, due to the difficulty
in solving the reduced equation or in evaluating the necessary quadrature to
recover the solution to the initial equation.

######  Example 4.1.

Consider the second-order ODE

$u_{2}=-\dfrac{(u_{1}+1)(x+u)}{xu},\quad xu\neq 0,$ (33)

which admits only the Lie point symmetry

$\mathbf{v}=x\partial_{x}+u\partial_{u}.$ (34)

The classical order-reduction method leads to the Abel-type equation

$w_{y}={\frac{\left(y+1\right)^{2}}{y}}w^{3}+{\frac{\left(2\,y+1\right)}{y}w^{2}},$
(35)

where

$y={\dfrac{u}{x}}\quad\mbox{and}\quad w={\dfrac{x}{u_{1}x-u}}.$ (36)

Maple system provides a rather involved implicit solution of (35) in terms of
the exponential integral function, from which it does not seem possible to
recover the general solution for (33).

In order to find a $\mathcal{C}^{\infty}$-structure for equation (33) we first
seek for a $\mathcal{C}^{\infty}$-symmetry of the form
$(\partial_{u},\lambda).$ The solution $\lambda=\dfrac{u_{1}+1}{u}$ for the
corresponding determining equation (31) can be easily determined. Therefore we
can choose as the first element of the $\mathcal{C}^{\infty}$-structure the
vector field

$X_{1}=(\partial_{u})^{[\lambda,(1)]}=\partial_{u}+\dfrac{u_{1}+1}{u}\partial_{u_{1}},$

which satisfies $[X_{1},A]=\lambda X_{1}.$ The corresponding form (30) becomes

$\omega_{2}=X_{1}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}=\dfrac{(u_{1}+1)(x+u+xu_{1})}{xu}dx-\dfrac{u_{1}+1}{u}du+du_{1}.$

According to Theorem 4.1 the Pfaffian equation $\omega_{2}\equiv 0$ is
completely integrable.

In order to integrate $\omega_{2}\equiv 0$, we first observe that, for the
admitted Lie point symmetry (34), we have that
$V=\mathbf{v}^{(1)}=x\partial_{x}+u\partial_{u}$ satisfies

$[V,X_{1}]=-X_{1},$

i.e., $V$ is a symmetry of the distribution $\mathcal{S}(\\{A,X_{1}\\}).$ In
this situation, according to Theorem 2.1 in [13], it is possible to use the
symmetry $V$ to construct directly an integrating factor of $\omega_{2}.$ Such
integrating factor is given by

$\mu_{2}(x,u,u_{1})=\dfrac{1}{V\,\lrcorner\,\omega_{2}}=\dfrac{u}{x(1+u_{1})^{2}}$
(37)

and a corresponding primitive for the exact 1-form $\mu_{2}\omega_{2}$ arises
by a simple quadrature:

$I_{2}(x,u,u_{1})=\ln(x)-\dfrac{u}{x(1+u_{1})}.$

We can choose $X_{2}=\partial_{u_{1}}$ as the second element of a
$\mathcal{C}^{\infty}$-structure for the equation, because
$\\{A,X_{1},X_{2}\\}$ are pointwise linearly independent (see Remark 3.2 (d)).
Indeed, it can be checked that the following commutation relations hold:

$[X_{1},A]=\lambda
X_{1},\qquad[X_{2},A]=X_{1}-\left(\dfrac{u_{1}x+2x+u}{xu}\right)X_{2},\qquad[X_{2},X_{1}]=\dfrac{1}{u}X_{2}.$

Observe that neither $X_{1}$ nor $X_{2}$ are symmetries of
$\mathcal{S}(\\{A\\})$ and that $X_{2}$ is not a symmetry of
$\mathcal{S}(\\{A,X_{1}\\}).$

The second completely integrable Pfaffian equation associated to the
$\mathcal{C}^{\infty}$-structure $\langle X_{1},X_{2}\rangle$ is defined by
the restriction of
$\omega_{1}=X_{2}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}=u_{1}dx-du$ to
the hypersurface defined by $I_{2}(x,u,u_{1})=C_{2},$ $C_{2}\in\mathbb{R}.$
Such hypersurface can be parametrized through

$\iota_{2}:(x,u)\mapsto\left(x,u,\dfrac{u}{x(\ln(x)-C_{2})}-1\right)$

and therefore

$\bar{\omega}_{1}:=\iota_{2}^{*}(\omega_{1})=\left(\dfrac{u}{x(\ln(x)-C_{2})}-1\right)dx-
du.$

The 1-form $\bar{\omega}_{1}$ is completely integrable, although is not
closed. However, it is easy to find an integrating factor that only depends on
$x:$

$\mu_{1}(x;C_{2})=\frac{1}{\ln(x)-C_{2}}.$ (38)

A corresponding primitive arises by a simple quadrature:

$I_{1}(x,u;C_{2})=\frac{u}{C_{2}-\ln(x)}+\phi(x;C_{2}),$

where

$\phi^{\prime}(x;C_{2})=\frac{1}{C_{2}-\ln(x)},$

and the prime denotes derivation with respect to $x.$

Consequently, if we solve for $u$ in $I_{1}(x,u;C_{2})=C_{1}$, where
$C_{1}\in\mathbb{R}$, we obtain the explicit solution of (33) in the form:

$u(x)=(\ln(x)-C_{2})(\phi(x;C_{2})-C_{1}),\qquad C_{1},C_{2}\in\mathbb{R}.$

It is worth noting that the general solution of the Abel equation (35) can be
obtained as by-product of our procedure. By using (36), the general solution
of the Abel equation (35) can be expressed in parametric form as follows:

$\begin{array}[]{lll}y(x)&=&{\dfrac{(\ln(x)-C_{2})(\phi(x;C_{2})-C_{1})}{x}},\\\
w(x)&=&-{\dfrac{x}{\left(\ln(x)-C_{2}\right)\left(\phi(x;C_{2})-C_{1}\right)-\phi(x;C_{2})+C_{1}+x}}.\end{array}$

In the following example we apply the $\mathcal{C}^{\infty}$-structure method
to completely solve a second-order ODE lacking Lie point symmetries.

######  Example 4.2.

It can be checked that the second-order ODE

$u_{2}+\dfrac{x(u_{1}+x)}{u}+\dfrac{u}{x^{4}}+1=0,\quad xu\neq 0$ (39)

does not admit Lie point symmetries. However, it is not difficult to find a
particular solution of the form $\lambda=a(x,u)u_{1}+b(x,u)$ to the
corresponding determining equation (31), giving rise to the
$\mathcal{C}^{\infty}$-symmetry $(\partial_{u},\lambda)$ defined by the
function $\lambda=\dfrac{u_{1}+x}{u}.$ Therefore

$X_{1}=(\partial_{u})^{[\lambda,(1)]}=\partial_{u}+\dfrac{u_{1}+x}{u}\partial_{u_{1}}$

can be chosen as the first vector field of a $\mathcal{C}^{\infty}$-structure
for (39).

The corresponding 1-form
$\omega_{2}=X_{1}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}$ becomes

$\omega_{2}=\left(\dfrac{(u_{1}+x)^{2}}{u}+\dfrac{u}{x^{4}}+1\right)dx-\dfrac{u_{1}+x}{u}du+du_{1}.$
(40)

The condition $dI_{2}\wedge\omega_{2}=0$ provides a determining system of
partial differential equations for a first integral $I_{2}$ of
$\mathcal{S}(\\{\omega_{2}\\}),$ where the subscripts denote the corresponding
partial derivatives:

$\displaystyle(I_{2})_{u}+\left(\dfrac{x+u_{1}}{u}\right)\,(I_{2})_{u_{1}}$
$\displaystyle=0,$ (41)
$\displaystyle(I_{2})_{x}+\left(\dfrac{(u_{1}+x)^{2}}{u}+\dfrac{u}{x^{4}}+1\right)\,(I_{2})_{u_{1}}$
$\displaystyle=0.$ (42)

The general solution to equation (41) becomes

$I_{2}(x,u,u_{1})=F\left(x,\dfrac{x+u_{1}}{u}\right),$ (43)

where $F=F(x,t)$ is an arbitrary smooth function. From equation (42) it
follows that the function $F=F(x,t)$ must satisfy:

$\left(\dfrac{1}{x^{4}}+t^{2}\right)F_{t}-F_{x}=0,$ (44)

which admits the particular solution
$F(x,t)=\arctan\left({tx^{2}-x}\right)-\dfrac{1}{x}.$ Therefore, by (43), the
function

$I_{2}(x,u,u_{1})=\arctan\left(\dfrac{x(xu_{1}+x^{2}-u)}{u}\right)-\dfrac{1}{x}$
(45)

is a first integral of $\mathcal{S}(\\{\omega_{2}\\}).$ Therefore
$\Sigma_{2}:=\\{(x,u,u_{1})\in M^{(1)}:I_{2}(x,u,u_{1})=C_{2}\\},$ where
$C_{2}\in\mathbb{R},$ is an integral manifold of $\mathcal{S}(\\{A,X_{1}\\}).$

In order to complete the $\mathcal{C}^{\infty}$-structure, let us recall that
by Remark 3.2 (d), $X_{2}$ may be any vector field such that
$\\{A,X_{1},X_{2}\\}$ is a set of pointwise linearly independent vector
fields. If we choose the vector field $X_{2}=\partial_{u_{1}}$ then
$\omega_{1}=u_{1}dx-du.$

We use the parametrization

$\iota_{2}(x,u)=\left(x,u,\dfrac{u}{x^{2}}{\tan\left(C_{2}+\dfrac{1}{x}\right)}-x+\dfrac{u}{x}\right)$

of the integral manifold $\Sigma_{2}$ to calculate

$\bar{\omega}_{1}:=\iota_{2}^{*}(\omega_{1})=\left(\dfrac{u}{x^{2}}{\tan\left(C_{2}+\dfrac{1}{x}\right)}-x+\dfrac{u}{x}\right)dx-
du.$ (46)

Although (46) is not closed, it is easy to calculate an integrating factor
that only depends on $x$:

$\mu_{1}(x;C_{2})=\dfrac{1}{x\cos\left(C_{2}+\dfrac{1}{x}\right)}.$ (47)

A primitive for the exact form $\mu_{1}\bar{\omega}_{1}$ becomes

$I_{1}(x,u;C_{2})=-\dfrac{u}{x\cos\left(C_{2}+\dfrac{1}{x}\right)}+\psi(x;C_{2}),$
(48)

where $\psi(x;C_{2})$ satisfies

$\psi^{\prime}(x;C_{2})=-\dfrac{1}{\cos\left(C_{2}+\dfrac{1}{x}\right)}.$

From the implicit solution $I_{1}(x,u;C_{2})=C_{1},$ $C_{1}\in\mathbb{R},$ we
finally obtain the explicit solution for equation (39) in the form

$u(x)=x\cos\left(C_{2}+\dfrac{1}{x}\right)\left(\psi(x;C_{2})-C_{1}\right),\quad
C_{1},C_{2}\in\mathbb{R}.$ (49)

###  4.3 $\mathcal{C}^{\infty}$-structures for third-order equations: an
example

In the following example we consider a third-order equation which admits just
one Lie point symmetry. The associated second-order reduced equation does not
admit Lie point symmetries. We first construct a
$\mathcal{C}^{\infty}$-structure of the third-order ODE and show how it can be
used to integrate the equation completely.

######  Example 4.3.

The third-order ODE

$u_{3}=\frac{xu_{2}e^{x}+2u_{1}e^{x}-3u_{2}e^{x}+3u_{2}^{2}}{xe^{x}+2u_{1}}$
(50)

only admits the Lie point symmetry $\partial_{u}$, which provides, by means of
the transformation $y=x$ and $w=u_{1},$ the obvious order reduction

$w_{2}=\dfrac{yw_{1}e^{y}+2we^{y}-3w_{1}e^{y}+3w_{1}^{2}}{ye^{y}+2w}.$ (51)

It can be checked that equation (51) does not admit Lie point symmetries, and
consequently, the classical Lie method cannot be applied to solve equation
(50).

In order to construct a $\mathcal{C}^{\infty}$-structure for equation (50) we
can use the Lie point symmetry $\partial_{u}$ to get the first element
$X_{1}:=(\partial_{u})^{(2)}=\partial_{u},$ for which $[X_{1},A]=0$ (i.e., the
functions $\lambda^{1}$ and $c^{1}$ in (23) are identically null).

For a second element of the $\mathcal{C}^{\infty}$-structure, we can consider
a vector field of the form

$X_{2}=\partial_{u_{1}}+\eta(x,u_{1},u_{2})\partial_{u_{2}},$

which satisfies $[X_{1},X_{2}]=0$ (i.e., the functions
$\gamma_{1}^{2},\bar{d}_{1}^{2}$ and $\bar{d}_{1}^{2}$ in (24) are identically
null). The determining equation for the function $\eta$ results from the
condition that the distribution $\mathcal{S}(\left\\{A,X_{1},X_{2}\right\\})$
must be involutive. By Proposition 2.1 this condition can be formulated in
terms of corresponding 1-form $\omega_{3}$ defined in (26) as
$\omega_{3}\wedge d\omega_{3}=0.$ For simplicity we search for a particular
solution of the determining equation of the form
$\eta(x,u_{1},u_{2})=a(x,u_{1})u_{2}+b(x,u_{1}),$ giving rise to

$\eta(x,u_{1},u_{2})=\displaystyle\frac{2(u_{2}-e^{x})}{xe^{x}+2u_{1}}.$

The last element $X_{3}$ of a $\mathcal{C}^{\infty}$-structure can be any
vector field such that the vector fields $A,X_{1},X_{2}$ and $X_{3}$ are
pointwise linearly independent (see Remark 3.2 (d)). Therefore, the ordered
set $\langle X_{1},X_{2},X_{3}\rangle$ defined by

$X_{1}=\partial_{u},\quad
X_{2}=\partial_{u_{1}}+\dfrac{2(u_{2}-e^{x})}{xe^{x}+2u_{1}}\partial_{u_{2}},\quad
X_{3}=\partial_{u_{2}}$

is a $\mathcal{C}^{\infty}$-structure for equation (50). This can be checked
by simply computing the corresponding Lie brackets.

Once a $\mathcal{C}^{\infty}$-structure has been determined, we can use the
algorithm described in Section 4 to integrate the given equation. Firstly, we
calculate the corresponding 1-forms (26), which in this example become:

$\begin{array}[]{l}\omega_{1}=-u_{1}dx+du,\quad\omega_{2}=u_{2}dx-du_{1},\\\
\\\
\omega_{3}=-\displaystyle\frac{(xu_{2}e^{x}+2u_{1}e^{x}-u_{2}e^{x}+u_{2}^{2})}{xe^{x}+2u_{1}}dx+\displaystyle\frac{2(e^{x}-u_{2})}{xe^{x}+2u_{1}}du_{1}+du_{2}.\end{array}$

According to Theorem 3.1 the Pfaffian equation $\omega_{3}\equiv 0$ is
completely integrable. The condition $dI_{3}\wedge\omega_{3}=0$ provides the
following system of determining equations for a first integral $I_{3}$ of
$\mathcal{S}(\\{\omega_{3}\\}):$

$\displaystyle(I_{3})_{u}$ $\displaystyle=0,$ (52)
$\displaystyle(xe^{x}+2u_{1})(I_{3})_{u_{1}}+2(u_{2}-e^{x})(I_{3})_{u_{2}}$
$\displaystyle=0,$ (53)
$\displaystyle(xe^{x}+2u_{1})(I_{3})_{x}+(u_{2}^{2}+xe^{x}u_{2}+2u_{1}e^{x}-e^{x}u_{2})(I_{3})_{u_{2}}$
$\displaystyle=0.$ (54)

It follows from equations (52) and (53) that

$I_{3}=F\left(x,\dfrac{u_{2}-e^{x}}{e^{x}+2u_{1}}\right),$ (55)

where $F=F(x,t)$ is, in principle, an arbitrary smooth function. By equation
(54) the function $F=F(x,t)$ must satisfy

$t^{2}F_{t}+F_{x}=0,$

whose general solution becomes $F(x,t)=G\left(x+\dfrac{1}{t}\right).$ In
consequence, by (55), a particular solution to (52)-(54) is given by:

$I_{3}(x,u,u_{1},u_{2})=\displaystyle\frac{xu_{2}+2u_{1}}{u_{2}-e^{x}}.$ (56)

The next step in our procedure is to restrict $\omega_{1}$ and $\omega_{2}$ to
the manifold implicitly defined by $I_{3}(x,u,u_{1},u_{2})=C_{3},$ by defining
the imbbeding

$\iota_{3}:(x,u,u_{1})\mapsto\left(x,u,u_{1},\frac{C_{3}e^{x}+2u_{1}}{C_{3}-x}\right).$

We obtain

$\begin{array}[]{l}\bar{\omega}_{1}=\iota_{3}^{*}(\omega_{1})=-u_{1}dx+du,\quad\bar{\omega}_{2}=\iota_{3}^{*}(\omega_{2})=\displaystyle\frac{C_{3}e^{x}+2u_{1}}{C_{3}-x}dx-
du_{1}.\\\ \end{array}$

According to Theorem 3.1 the Pfaffian equation $\bar{\omega}_{2}\equiv 0$ is
completely integrable, i.e., $\bar{\omega}_{2}\wedge d\bar{\omega}_{2}=0.$ In
this case an integrating factor only depending on $x$ can be easily found:

$\mu_{2}(x;C_{3})=-(C_{3}-x)^{2}.$

A corresponding primitive of $\mu_{2}\bar{\omega}_{2}$ can be calculated by
quadrature and it is given by

$I_{2}(x,u,u_{1};C_{3})=C_{3}e^{x}(x-C_{3}-1)+u_{1}(x-C_{3})^{2}.$

In order to restrict $\bar{\omega}_{1}$ to $I_{2}(x,u,u_{1};C_{3})=C_{2},$ we
solve for $u_{1}$ in this last equation and define

$\iota_{2}:(x,u)\mapsto\left(x,u,\dfrac{C_{3}e^{x}(C_{3}-x+1)+C_{2}}{(C_{3}-x)^{2}}\right).$

The pullback of $\bar{\omega}_{1}$ by $\iota_{2}$ becomes

$\widetilde{\omega}_{1}:=\iota_{2}^{*}(\bar{\omega}_{1})=-\dfrac{C_{3}e^{x}(C_{3}-x+1)+C_{2}}{(C_{3}-x)^{2}}dx+du.$

In this case the 1-form $\widetilde{\omega}_{1}$ is not only Frobenius
integrable, but closed, i.e., $d\widetilde{\omega}_{1}=0.$ Therefore the
1-form $\widetilde{\omega}_{1}$ is locally exact and a corresponding primitive
can be computed by simple quadrature:

$I_{1}(x,u;C_{3},C_{2})=\displaystyle\frac{C_{3}(e^{x}-u)+ux+C_{2}}{x-C_{3}}.$

Finally, from the implicit solution $I_{1}(x,u;C_{3},C_{2})=C_{1}$ we can
obtain the exact general solution of equation (50) in explicit form:

$u(x)=\dfrac{C_{1}(C_{3}-x)+C_{3}e^{x}+C_{2}}{C_{3}-x},\qquad
C_{1},C_{2},C_{3}\in\mathbb{R}.$

It is interesting to observe that, as by-product, our procedure also provides
the general solution of the second-order equation (51):

$w(y)=\dfrac{C_{2}-C_{3}e^{y}(y-C_{3}-1)}{(y-C_{3})^{2}}.$

Recall that equation (51) does not admit Lie point symmetries and, as far as a
we know, cannot be solved by standard procedures.

###  4.4 $\mathcal{C}^{\infty}$-structures for systems of ODEs

In this section we show how the results that have been presented in Section
4.1 for scalar ODEs can be adapted and applied to systems of ODEs.

It is well-known that by introducing auxiliary dependent variables, if that is
necessary, any system of ODEs can be transformed into a system of first-order
ODEs. We also assume that the system can be written in the normal form:

$u^{i}_{1}=F^{j}(x,u^{1},\ldots,u^{r}),\quad 1\leq i\leq r.$ (57)

We consider the vector field associated to system (57):

$A=\partial_{x}+\displaystyle\sum_{i=1}^{r}F^{i}(x,u^{1},\ldots,u^{r}){\partial}_{u^{i}}.$
(58)

A $\mathcal{C}^{\infty}$-structure for the distribution generated by $A,$ in
the sense of Definition 2.2, will be called a $\mathcal{C}^{\infty}$-structure
for (57). In this case, a $\mathcal{C}^{\infty}$-structure is an ordered
collection of $r$ smooth vector fields $\langle X_{1},\ldots,X_{r}\rangle$ on
some open set $M$ of $\mathbb{R}^{r+1}$ that must satisfy relationships of the
form (23)-(25), where now $A$ is given by (58). The last element $X_{r},$ as
in the case of scalar ODEs, may be any vector field such that
$\\{A,X_{1},\dots,X_{r}\\}$ is a set of pointwise linearly independent vector
fields on $M.$ A $\mathcal{S}(\\{A\\})$-related
$\mathcal{C}^{\infty}$-structure of 1-forms for system (57) can be explicitly
constructed by using the volume form $\boldsymbol{\Omega}=dx\wedge
du^{1}\wedge\ldots\wedge du^{r}:$

$\begin{array}[]{lll}\omega_{i}&=&{X_{1}\,\lrcorner\,X_{2}\,\lrcorner\,\ldots\,\lrcorner\,\widehat{X_{i}}\,\lrcorner\,\ldots\,\lrcorner\,X_{r}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}},\quad
1\leq i\leq r.\end{array}$ (59)

Such $\mathcal{C}^{\infty}$-structure of 1-forms permits to solve the system
through the integration of $r$ successive completely integrable Pfaffian
equations, as stated in Theorem 3.1. This procedure is illustrated in the
following example.

######  Example 4.4.

Consider the following system

$\left\\{\begin{array}[]{l}\dfrac{dx(t)}{dt}=\dfrac{ty+x-2y\tan(t)}{t-\tan(t)},\\\
\dfrac{dy(t)}{dt}=\dfrac{x+(tx-t^{2}y-y)\tan(t)}{t^{2}-t\tan(t)}\end{array}\right.$
(60)

and denote by $A$ its associated vector field (58).

The first element of a $\mathcal{C}^{\infty}$-structure for
$\mathcal{S}(\\{A\\})$ can be searched, for simplicity, as a vector field of
the form $X_{1}=\partial_{x}+\eta(t)\partial_{y}.$ The condition
$[X_{1},A]\in\mathcal{S}(\mathcal{\\{}A,X_{1}\\})$ can be expressed in terms
of the corresponding 1-form
$\omega_{2}=X_{1}\,\lrcorner\,A\,\lrcorner\,\boldsymbol{\Omega}$ given in (59)
as $\omega_{2}\wedge d\omega_{2}=0.$ This provides the following determining
equation for $\eta(t):$

$\eta^{\prime}(t)=\frac{2\tan(t)-t}{t-\tan(t)}\eta(t)^{2}-\frac{t^{2}\tan(t)+t+\tan(t)}{t^{2}-t\tan(t)}\eta(t)+\frac{t\tan(t)+1}{t^{2}-t\tan(t)},$

which admits the particular solution $\eta(t)=1/t$.

The second element of the $\mathcal{C}^{\infty}$-structure can be any vector
field pointwise linearly independent with $\\{A,X_{1}\\}$ (Remark 3.2 (d)). We
can choose, for instance, $X_{2}=\partial_{y}.$ Therefore, the ordered set
$\langle X_{1},X_{2}\rangle$ defined by the vector fields

$X_{1}={\partial_{x}}+{\dfrac{1}{t}\partial_{y}},\quad X_{2}=\partial_{y}$

constitutes a $\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{A\\})$.

A $\mathcal{S}(\\{A\\})$-related $\mathcal{C}^{\infty}$-structure of 1-forms
(i.e., satisfying conditions (16)), is given by the differential forms
corresponding to (59)

$\begin{array}[]{l}\omega_{1}=\dfrac{yt+x-2y\tan(t)}{t-\tan(t)}dt-dx,\\\
\omega_{2}=\dfrac{yt^{2}\tan(t)+yt-
xt\tan(t)-y\tan(t)}{t^{2}-t\tan(t)}dt-\dfrac{1}{t}dx+dy.\end{array}$ (61)

By Theorem 3, the Pfaffian equation $\omega_{2}\equiv 0$ is completely
integrable. In order to find a first integral of
$\mathcal{S}(\\{\omega_{2}\\})$ we use the condition
$dI_{2}\wedge\omega_{2}=0$ and obtain:

$\displaystyle(I_{2})_{y}+t(I_{2})_{x}=0,$ (62)
$\displaystyle\left(yt^{2}\tan(t)+yt-
xt\tan(t)-y\tan(t)\right)(I_{2})_{x}+\left(t-\tan(t)\right)(I_{2})_{t}=0.$
(63)

From (62) it follows that

$I_{2}(t,x,y)=F\left(t,\dfrac{yt-x}{t}\right),$

where $F(t,s)$ is, in principle, an arbitrary smooth function. By using (63)
we conclude that $F$ must satisfy the partial differential equation

$s\left(\tan(t)-\tan(t)t^{2}-t\right)F_{s}+t(t-\tan(t))F_{t}=0,$ (64)

whose general solution becomes

$F(t,s)=G\left(\frac{ts}{t\cos(t)-\sin(t)}\right),$

where $G$ is an arbitrary smooth function. Therefore, a particular solution
for (62)-(63) is

$I_{2}(t,x,y)=\dfrac{yt-x}{t\cos(t)-\sin(t)}.$ (65)

The level set given by $I_{2}(t,x,y)=C_{2},$ with $C_{2}\in\mathbb{R},$
defines an integral manifold $\Sigma_{2}$ of the distribution
$\mathcal{S}(\\{A,X_{1}\\}).$ A local parametrization is defined by the
imbedding

$\iota_{2}:(t,x)\mapsto\left(t,x,\dfrac{C_{2}t\cos(t)-C_{2}\sin(t)-x}{t}\right).$

The restriction of the 1-forms in (61) to $\Sigma_{2}$ become
$\iota_{2}^{*}(\omega_{2})=0$ (because $\Sigma_{2}$ is an integral manifold of
$\mathcal{S}(\\{\omega_{2}\\})$) and

$\bar{\omega}_{1}:=\iota_{2}^{*}(\omega_{1})=\dfrac{C_{2}t\cos(t)-2C_{2}\sin(t)+2x}{t}dt-
dx.$

The Pfaffian equation $\bar{\omega}_{1}\equiv 0$ is also completely
integrable. The condition $dI_{1}\wedge\bar{\omega}_{1}=0$ gives rise to the
equation

$\left(2C_{2}\sin(t)-C_{2}t\cos(t)-2x\right)(I_{1})_{x}-t(I_{1})_{t}=0,$ (66)

which admits the particular solution

$I_{1}(t,x,y;C_{2})=\dfrac{x-C_{2}\sin(t)}{t^{2}}.$ (67)

This function is a first integral of
$\bar{A}=(\iota_{1}^{-1})_{*}\left({A|}_{{\Sigma}_{2}}\right).$ Therefore
$I_{1}(t,x,y;C_{2})=C_{1}$, with $C_{1}\in\mathbb{R}$, defines an integral
manifold $\bar{\Sigma}_{1}$ of $\mathcal{S}(\bar{A}),$ which can be
parametrized through the map

$\iota_{1}:t\mapsto(t,C_{1}t^{2}+C_{2}\sin(t)).$

Finally, $\Sigma:=\iota_{2}(\bar{\Sigma}_{1})$ is an integral manifold of
$\mathcal{S}(\\{A\\})$ that can be parametrized by
$\iota:=\iota_{2}\circ\iota_{1}:$

$\iota:t\mapsto(t,C_{1}t^{2}+C_{2}\sin(t),C_{2}\cos(t)+C_{1}t).$

Therefore,

$x(t)=C_{1}t^{2}+C_{2}\sin(t),\,y(t)=C_{2}\cos(t)+C_{1}t,$

where $C_{1},C_{2}\in\mathbb{R},$ is the general solution of system (60).

Observe that in this example, the first integral $I_{2}$ in (65) associated to
$\omega_{2}\equiv 0$ has been obtained by solving the PDEs (62) and (64),
while the first integral $I_{1}$ in (67) associated to $\omega_{2}\equiv 0$
has been obtained from equation (66). It can be checked that the
characteristic systems associated to these three PDEs correspond to first-
order ODEs that are linear, and they are therefore easy to integrate. This
shows that, although the Pfaffian equations associated to a
$\mathcal{C}^{\infty}$-structure might not be defined by closed 1-forms (as in
the case of solvable structures), there are examples, as in this case, in
which such Pfaffian equations are simple and can be solved by standard
procedures.

##  5 Discussion and related approaches

For any involutive distribution $\mathcal{Z}$ we have introduced a new
structure of vector fields, called $\mathcal{C}^{\infty}$-structure, which
plays an outstanding role in the determination of the integral manifolds of
$\mathcal{Z}$ (Theorem 3.1). For the particular case where $\mathcal{Z}$ is
defined by a single vector field associated with an ODEs system, the new
approach greatly enlarges the types of vector fields that have been considered
in the last decades with the ultimate goal of integrating or reducing the
underlying differential equations.

In this section we present a short review and synthesis of some relevant
aspects of approaches to that goal. First of all, it should be clear that
these theories have a long history whose origins can be traced back to the
works of F. G. Frobenius and S. Lie. The main ideas have been gradually
evolving over the years by following different objectives and motivations.
Some of the Lie’s ideas derived to an algebraic synthesis that led to a new
branch of Mathematics (the algebraic Lie theory) whose objectives are very
different to the original ones.

The geometric theory of differential equations emerged thanks to the works of
S. Lie, E. Cartan and others in the first half of the 20th century. The
methods in this theory are somewhat different from the classical (analytical)
methods. This makes it difficult an exhaustive comparison and better
understanding of the achievements in both theories (analytical and
geometrical). This would be, undoubtedly, an arduous task, because such
approaches usually emerge from different motivations, pursue diverse
objectives and, hence, are presented and investigated from different
perspectives, by using different mathematical languages and notations.
Nevertheless, we believe that these theories are mutually imbricated and,
despite their differences, any of them can offer a very positive influence on
the others. In other words, any effort to clarify, reinterpret, combine and
complement the new results has certainly beneficial effects to the different
approaches.

In the context of the present paper, one of the most important concepts that
extend the notion of solvable algebra of local symmetries is the idea of
solvable structure. According to P. Basarab-Horwath [8], the origins of the
concept of solvable structure (in the scheme of Frobenius approach) can be
traced back to the works of Sophus Lie. At the begining of the 90s, thanks to
the works of P. Basarab-Horwath [8], J. Sherring and G. E. Prince [13], T.
Hartl and C. Athorne [9], M.A. Barco and G.E. Prince [14], and others, the
theory of solvable structures received new impulses and inspired many
developments from both analytical and geometrical point of views.

With the objective of extending Lie reduction methods to find exact solutions
of equations not possessing Lie point symmetries, the concept of
$\lambda$-prolongation, and its characterization in terms of the invariants-
by-derivation property (IBDP) were introduced at the beginning of the twenty-
first century [10, 28]. E. Pucci and G. Saccomandi provided in [29] an
interesting geometric interpretation of these prolongations and introduced the
so-called telescopic vector fields. In 2007 D. Catalano-Ferraioli proved that
these prolongations can be interpreted as shadows of some nonlocal symmetries,
by using the technique of the $\lambda$-coverings [30]. Along the first decade
of this century, G. Gaeta, P. Morando, G. Cicogna and S. Walcher explored
several methods of prolongation of vector fields that generalized the concept
of $\lambda$-prolongation. Collectively, these prolongations were called
twisted prolongations (see the review papers [24] and [31] by G. Gaeta and
references therein). Depending on the context where the prolongations are
being considered, these generalizations and their corresponding notions of
symmetry, received specific names, according to the auxiliary object they use.
Thus, for instance, the auxiliary object for $\lambda$-symmetries of an ODE,
is a scalar function $\lambda$ on the jet space; for $\mu$-symmetries of a
PDE, the auxiliary object is a semi-basic one-form $\mu=\Lambda_{i}dx^{i}$,
where the $\Lambda_{i}$ are matrix functions, satisfying the Maurer-Cartan
equation [20, 25, 32]. Similar extensions were carried out for dynamical
systems, and specifically for Hamiltonian systems, [33, 34] and for
variational problems [21, 35, 36]. A review on these issues appears in [23].

In 2012, G. Cicogna, G. Gaeta and S. Walcher introduced the concept of
$\sigma$-symmetry in the study of the integrability of systems of ODEs [22].
These authors, in 2013, used $\sigma$-symmetries to separately study dynamical
systems [34] and orbital reducibility [37]. The initial concept is based on
the notion of $\sigma$-prolongation (also called joint $\lambda$-prolongation)
of a set of $r$ vector fields defined on the 0th-order jet space $M$ [22,
Definition 2]. In this case, the auxiliary object is a $r\times r$ matrix
whose coefficients are smooth functions $\sigma^{i}_{j}$ on $M^{(1)}.$ An
important aspect about $\sigma$-prolonged vector fields is that, as in the
case of $\lambda$-prolongations, they maintain the IBDP and therefore the
$\sigma$-symmetries are as useful as standard ones for reduction purposes.
This fact is the key to adapt the original Lie symmetry (or the
$\lambda$-symmetry) reduction method, i.e., to reduce a system of ODEs which
is invariant under a set of $\sigma$-prolonged vector fields. However, as it
is remarked in [22, Section 3], the IBDP for $\sigma$-prolonged vector fields,
and hence the associated reduction procedure, require that the
$\sigma$-prolonged vector fields must constitute an involutive system.
Assuming that the base vector fields are in involution, the authors provide a
sufficient condition for the $\sigma$\- prolongations be in involution,
although the commutation relationships may be in general different from the
original ones.

Among the questions brought forward in [22, Section 6], the authors propose
the possible extensions of the $\sigma$-symmetry theory to the study of Euler-
Lagrange equations and to the study of nonlocal symmetries, in the line of the
works by D. Catalano-Ferraioli and P. Morando on solvable structures for ODEs
[38, 16]. These authors adapted solvable structures to local and nonlocal
symmetries (classical and higher), introducing the notion of nonlocal solvable
structure. For the problem of determining solvable structures of certain
covering system, which in practice is usually a difficult one, the authors
observed that previously known symmetry generators can help to find them.

The authors of [22] did also encourage the investigation of the use of
$\sigma$-symmetries for a possible generalization of solvable structures. As
far as we know, this is still an open problem. Observe that, in some sense,
the results in this paper on $\mathcal{C}^{\infty}$-structures are aimed in
that direction, although strictly speaking they are not based on
$\sigma$-symmetries, as suggested in [22], but on
$\mathcal{C}^{\infty}$-symmetries of distributions.

In [39], P. Morando presented a dimensional reduction method for a vector
field $Z$ defined on an $n$-dimensional manifold, based on the knowledge of a
suitable integrable distribution $\mathcal{D}_{0}$ of $r<n$ vector fields. The
reconstruction of the general solution to the system associated with $Z$ from
the solutions of the reduced system was also analyzed. In particular, if
$\mathcal{D}_{0}$ is a solvable structure for $Z$, then the integral lines of
$Z$ can be reconstructed by quadrature from the reduced ones. This kind of
distributions are named in [39] partial solvable structures. For the case of
systems of ODEs, the reduction procedure based on $\sigma$-symmetries fits in
with this general reduction method. In the particular case in which the
$\sigma$-symmetry is a partial solvable structure (called a triangular
$\sigma$-symmetry), the reconstruction problem can be solved by quadratures.

In [40, 41] the integrability of any $\mbox{SL}(2,\mathbb{R})$-invariant
$n$th-order ODE was tackled by considering solvable structures with respect to
the involutive distribution formed by the vector field associated to the
equation and the symmetry generators of the nonsolvable symmetry algebra
$\mathfrak{sl}(2,\mathbb{R})$. This procedure was later extended for any
$n$th-order ODE with a known $k$-dimensional symmetry algebra
$\mathcal{G}_{k},$ solvable or not, with $1\leq k<n$ ([42]). These new
structures were named  generalized solvable structures with respect to
$\mathcal{G}_{k}.$

As remarked by G. Gaeta in [43], the above described theories show that the
focus must be on distributions rather than on a single vector field, and in
involution rather than in the Lie algebraic structure.

As a summary, some of the most significant differences between
$\mathcal{C}^{\infty}$-structures and the previous types of systems of vector
fields are highlighted below.

  1. 1.

The concepts of solvable algebra, solvable structure and
$\mathcal{C}^{\infty}$-structure are established for an involutive
distribution $\mathcal{Z}$ of $r$ vector fields on a $n$-dimensional manifold
through an ordered set of $n-r$ vector fields $\langle
X_{1},\ldots,X_{n-r}\rangle.$ For a solvable structure, as well as for a
solvable algebra, the first element $X_{1}$ must be a symmetry of
$\mathcal{Z};$ this condition is not required for a
$\mathcal{C}^{\infty}$-structure; in the commutation relationships between
$X_{1}$ and the fields in $\mathcal{Z}$ the vector field $X_{1}$ may appear.
Similar situations may occur for the successive elements.

  2. 2.

For a solvable algebra, the Lie bracket of two generators must be a linear
combination, with constant coefficients, of the generators. In particular, the
generators constitute an involutive system and have a complete system of joint
invariants. For solvable structures or $\mathcal{C}^{\infty}$-structures, that
is not the case. In the commutation relationships between an $X_{i}$ and an
$X_{i+j}$ there may appear the vector fields in $\mathcal{Z}$ and the
coefficients are, in general, non constant functions.

  3. 3.

The problem of the integration of the distribution $\mathcal{Z}$ by using
successive quadratures of locally exact one-forms can be solved either with
solvable algebras or with solvable structures. The results in this paper show
that with a $\mathcal{C}^{\infty}$-structure the integration problem can be
splitted into successive integrable Pfaffian equations, although not
necessarily integrable by quadratures.

  4. 4.

For both partial and generalized solvable structures, the involved vector
fields are still symmetries of certain distributions containing the vector
field associated to the equation. Nevertheless, it must be remarked that this
condition is not required for a $\mathcal{C}^{\infty}$-structure.

  5. 5.

As it has been said, for a $\mathcal{C}^{\infty}$-structure it is allowed that
the vector fields of the distribution $\mathcal{Z}$ may appear in the
commutation relationships. In particular, the vector fields in a
$\mathcal{C}^{\infty}$-structure not necessarily must be in involution. Hence,
the integration process carried out in this paper is not based on the IBDP (as
for $\sigma$-symmetries); in fact it is more related to the methods based on
solvable structures, as presented in [8, 9, 14, 15, 44, 45]. In consequence,
$\mathcal{C}^{\infty}$-structures permit to construct the integral manifolds
by solving successively $n-r$ completely integrable Pfaffian equations. At
each step, the space where each Pfaffian equation is defined has one less
dimension than in the previous stage.

##  6 Final remarks

Frobenius Theorem for involutive distributions of vector fields guarantees the
existence of integral manifolds through each point, although it does not
provide a constructive procedure to find them. Among the several concepts that
have been introduced in the literature to obtain constructive integration
methods, the concept of solvable structure is one of the most powerful. Once a
solvable structure has been determined, there is a systematic procedure to
integrate the underlying distribution by successive quadratures. The main
difficulty of this approach may be the determination of the elements of the
solvable structure. These elements must be symmetries of the successive
involutive distributions associated to the solvable structure.

In this work the concept of $\mathcal{C}^{\infty}$-symmetry, initially
introduced for ODEs, has been adapted for involutive distributions of vector
fields. This permits to extend the concept of solvable structure and introduce
the notion of $\mathcal{C}^{\infty}$-structure for an involutive distribution.

These new objects play a fundamental role in the integrability of the
distribution, with a somewhat similar functionality to
$\mathcal{C}^{\infty}$-symmetries in the integration of ODEs. In this paper it
has been proved that the knowledge of a $\mathcal{C}^{\infty}$-structure for a
given rank $r$ involutive distribution allows to compute their integral
manifolds by solving $n-r$ completely integrable Pfaffian equations. A
solvable structure is a particular case of a $\mathcal{C}^{\infty}$-structure.
Thus, $\mathcal{C}^{\infty}$-structures enlarge the classes of vector fields
that can be used to integrate involutive distributions of vector fields.

The main theoretical results have been applied to the particular case of
distributions generated by the vector fields associated to systems of ordinary
differential equations. In particular, it has been proved that a
$\mathcal{C}^{\infty}$-structure for a given $m$th-order scalar ODE leads to
the integration of the equation after solving successively $m$ completely
integrable Pfaffian equations. Each of these Pfaffian equations is defined in
an space with one less dimension than in the previous stage. The presented
examples show that, although these Pfaffian equations might not be defined by
closed 1-forms, as in the case of solvable structures, they can be equally
integrated. This has permitted to integrate completely the considered
equations, even when the classical Lie method fails.

The possible limitations in the application of the presented approach come
from the potential difficulty in finding particular solutions of the
determining equations for the calculation of the elements of a
$\mathcal{C}^{\infty}$-structure. In this regard, it would be very convenient
to investigate strategies in order to simplify such determining equations, as
it has been done in the literature for the search of
$\mathcal{C}^{\infty}$-symmetries of some specific form. Work in this line is
currently in progress.

## Acknowledgments

The authors thank the finantial support from Junta de Andalucía  to the
research group FQM–377. A. Ruiz and C. Muriel acknowledge the financial
support from FEDER–Ministerio de Ciencia, Innovación y Universidades–Agencia
Estatal de Investigación by means of the project PGC2018-101514-B-I00.

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