# Integration of differential equations by $\mathcal{C}^{\infty}$-structures
††thanks: Citation: Authors. Title. Pages…. DOI:000000/11111.

*A. J. Pan-Collantes, C. Muriel, A. Ruiz
Departamento de Matemáticas
Universidad de Cádiz - UCA
Puerto Real
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Several integrability problems of differential equations are addressed by
using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization
of the notion of solvable structure. Specifically, the integration procedure
associated with $\mathcal{C}^{\infty}$-structures is used to integrate to a
Lotka-Volterra model and several differential equations that lack sufficient
Lie point symmetries and cannot be solved using conventional methods.

##  1 Introduction

Solvable structures appeared in the last decade of the 20th century as a
generalization of the concept of solvable symmetry algebra [1, 2, 3, 4], in
order to characterize the integrability by quadratures of an involutive
distribution of vector fields $\mathcal{Z}$ on a $n$-dimensional manifold [5,
6, 7, 8]. Roughly speaking, a solvable structure for a distribution
$\mathcal{Z}$ of rank $r$ consists of a sequence of $n-r$ vector fields that
gives rise to a chain of distributions such that each vector field in the
structure is a symmetry of the previous distribution.

Almost at the same time, $\mathcal{C}^{\infty}$-symmetries were introduced as
a generalization of the classical Lie symmetry method of reduction [1, 2] for
ordinary differential equations (ODEs) [9]. Since its introduction
$\mathcal{C}^{\infty}$-symmetries have been extended in multiple directions
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and are being
extensively used [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39],
allowing to solve equations that may even lack Lie point symmetries [9, 40,
41].

The idea that allowed to extend the notion of Lie point symmetry to
$\mathcal{C}^{\infty}$-symmetry in the context of ODEs, has been adapted in
[42, 43] for involutive distributions of vector fields. The condition for a
vector field to be a $\mathcal{C}^{\infty}$-symmetry of a distribution is less
restrictive than for a symmetry, which implies that in practice the
$\mathcal{C}^{\infty}$-symmetries of a distribution are easier to find than
its symmetries. When in the notion of solvable structure we let the elements
be $\mathcal{C}^{\infty}$-symmetries, instead of symmetries, of the chains of
distributions mentioned above, we get a more general structure, which has been
called $\mathcal{C}^{\infty}$-structure in [42]. The key point in this new
theory is that once a $\mathcal{C}^{\infty}$-structure for an involutive
distribution $\mathcal{Z}$ of corank $k$ has been determined, then
$\mathcal{Z}$ can be integrated by sequentially solving $k$ integrable
Pfaffian equations [42, Theorem 3.5]. These Pfaffian equations are defined in
spaces whose dimensions decrease one unit at each stage. The Pfaffian
equations are completely integrable, although unlike solvable structures, they
may not be integrable by quadratures. The well-known result on the
relationship between integrating factors and Lie point symmetries for first-
order ODEs [1, 3, 4, 44] has been recently generalized in [43] for
$\mathcal{C}^{\infty}$-structures and involutive distributions of arbitrary
corank by introducing symmetrizing factors. Relevant results on the role of
these symmetrizing factors on the integrability by quadratures of the Pfaffian
equations that arise by application of the $\mathcal{C}^{\infty}$-structure
method have been also derived [43].

In this work we present some new applications of the integration procedure
associated to $\mathcal{C}^{\infty}$-structures. The paper is organized as
follows: in Sections 2 and 3 we recall the main definitions and results in the
theory of $\mathcal{C}^{\infty}$-structures, by adapting some of the
theoretical results that were obtained in [42] to the problems that we address
in this paper. In Section 4 we explore the application of the
$\mathcal{C}^{\infty}$-structure method to fully integrate two systems of
first-order ordinary differential equations, one of which is a Lotka-Volterra
system, frequently employed to describe the dynamics of biological systems.
Additionally, we investigate three scalar ODEs in Section 5, two of which are
of fourth order and one of third order. Notably, the considered equations
exhibit a lack of sufficient Lie point symmetries, and even powerful symbolic
systems like Maple fail to provide explicit solutions for them. Nevertheless,
utilizing our novel integration method based on
$\mathcal{C}^{\infty}$-structures, we achieve complete integration of the
equations that are difficult to solve by using conventional methods.

##  2 Preliminaries

In this paper functions, vector fields and differential forms are assumed to
be smooth (meaning $\mathcal{C}^{\infty}$) on a contractible open subset $U$
of $\mathbb{R}^{n}.$ In what follows, $\mathfrak{X}(U)$ and $\Omega^{k}(U)$
will denote the $\mathcal{C}^{\infty}(U)$-module of all smooth vectors and
$k$-forms, respectively, while $\Omega^{*}(U)$ will denote the exterior
differential algebra of all differential forms on $U.$

Given a set $\\{Z_{1},\ldots,Z_{r}\\}$ of pointwise linearly independent
vector fields on $U,$ by $\mathcal{Z}:=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$
we denote 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 pointwise linearly independent 1-forms
$\\{\sigma_{1},\ldots,\sigma_{s}\\}$ will be denoted by
$\mathcal{P}:=\mathcal{S}(\\{\sigma_{1},\ldots,\sigma_{s}\\}).$ The submodule
$\mathcal{Z}$ (resp. $\mathcal{P}$) defines a distribution (resp. a Pfaffian
system) of constant rank $n-r$ (resp. $n-s$).

The annihilator of $\mathcal{Z}$ is the set of the differential forms
$\omega\in\Omega^{*}(U)$ such that
$\omega(Y_{1},\ldots,Y_{k})=0$ whenever $Y_{1},\ldots,Y_{k}\in\mathcal{Z}.$
This set, which will be denoted by $\mbox{Ann}(\mathcal{Z}),$ is an ideal of
$\Omega^{*}(U)$ locally generated by $n-r$ pointwise linearly independent
1-forms $\\{\omega_{1},\ldots,\omega_{n-r}\\}$ [45, 46]. In this case, we will
write $\mathcal{Z}^{\circ}=\mathcal{S}(\\{\omega_{1},\ldots,\omega_{n-r}\\}).$
It can be checked that the Pfaffian system $\mathcal{Z}^{\circ}$ can be
characterized in terms of the interior product (or contraction) $\lrcorner$ as
follows:

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

Let us recall that the distribution $\mathcal{Z}$ is said to be involutive if
$[Z_{i},Z_{j}]\in\mathcal{Z}$ for $1\leq i,j\leq r.$ A well-known result
states that $\mathcal{Z}$ is involutive if and only if the ideal
$\mbox{Ann}(\mathcal{Z})$ is closed under exterior differentiation $d,$ i.e.,
if $\mbox{Ann}(\mathcal{Z})$ is a differential ideal (see, for instance,
Proposition 2.30 and Definition 2.29 in [45]). In this case, Frobenius Theorem
[45, Theorem 1.60] guarantees, for each $p\in U,$ the local existence of a
unique connected integral manifold of $\mathcal{Z}$ of maximal dimension [45,
Definition 1.63]. Such integral manifolds can be defined (locally) by the
level sets of a complete set of first integrals $I_{1},\ldots,I_{n-r}$ for the
distribution $\mathcal{Z}.$ It is clear that, in this case, the independent
1-forms $\\{dI_{1},\ldots,dI_{n-r}\\}$ generate the corresponding Pfaffian
system $\mathcal{Z}^{\circ},$ which is said to be completely integrable [45,
46]. In this sense, integrating a completely integrable Pfaffian system is
equivalent to integrating the corresponding involutive system of vector
fields.

In such integration procedures, the notion of solvable structure, introduced
by Basarab-Horwath in [5] plays a fundamental role (see also [7]). This
concept is based on the notion of symmetry of a distribution, which
generalizes Lie point symmetries: [47, 5, 48]:

######  Definition 2.1.

A symmetry of an involutive distribution $\mathcal{Z}$ is a vector field $X$
such that the set $\\{Z_{1},\ldots,Z_{r},X\\}$ is pointwise linearly
independent on $U$ and $[X,\mathcal{Z}]\subset\mathcal{Z}$.

Now we can recall the concept of solvable structure:

######  Definition 2.2.

[5, Definiton 4]: A solvable structure for $\mathcal{Z}$ consists of an
ordered set111Observe that in references such as [5, 6, 7, 8, 49, 50] the term
solvable structure refers to the whole ordered set $\langle
Z_{1},\ldots,Z_{r},X_{1},\ldots,X_{n-r}\rangle$ that also includes the vectors
fields that generate the distribution $\mathcal{Z}.$ of vector fields $\langle
X_{1},\ldots,X_{n-r}\rangle$ such that $X_{1}$ is a symmetry of $\mathcal{Z}$
and $X_{i}$ is a symmetry of the distribution
$\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{i-1}\\})$ for
$i=2,\ldots,n-r.$

The main result concerning solvable structures is that the knowledge of a
solvable structure allows us to find the integral manifolds of $\mathcal{Z}$,
at least locally, by quadratures alone [5, Proposition 3]. A dual version of
Definition 2.2, given in terms of differential 1-forms, was introduced in [6,
Defintion 4] by Hartl and Athorne. These authors also re-established the
integrability result by Basarab-Horwath from a dual point of view (see [6,
Proposition 5]). We refer the reader also to [8, 49] for further details on
the integration procedure associated to solvable structures.

Solvable structures are very useful in the study of ordinary differential
equations (ODEs), because such problems can be reformulated as the task of
integrating systems of vector fields or 1-forms. For instance, consider a
system of first-order ODEs

$\left\\{\begin{array}[]{lcl}\dot{x}_{1}&=&\phi_{1}(t,x_{1},\ldots,x_{n}),\\\
\dot{x}_{2}&=&\phi_{2}(t,x_{1},\ldots,x_{n}),\\\ &\vdots&\\\
\dot{x}_{n}&=&\phi_{n}(t,x_{1},\ldots,x_{n}),\\\ \end{array}\right.$ (1)

where $\phi_{1},\ldots,\phi_{n}$ are smooth functions on some open set
$U\subset\mathbb{R}^{n+1}$ and over dot denotes differentiation with respect
to the independent variable $t.$ Any solution of system (1) defines a one-
dimensional integral manifold of the (trivially involutive) rank 1
distribution generated by the vector field

$Z=\partial_{t}+\phi_{1}(t,x_{1},\ldots,x_{n})\partial_{x_{1}}+\ldots+\phi_{n}(t,x_{1},\ldots,x_{n})\partial_{x_{n}}.$
(2)

The extension to systems of ODEs of higher order is straightforward. Consider,
for instance, a general $m$th-order ODE:

$u_{m}=\phi(x,u^{(m-1)}),$ (3)

where $u^{(m-1)}=(u,u_{1},\ldots,u_{m-1})$ denotes the dependent variable $u$
and, for $1\leq k\leq m,$ $u_{k}$ denotes the derivative of order $k$ of $u$
with respect to the independent variable $x.$ By setting $x=t,$ $x_{1}=u,$ and
$x_{k}=u_{k-1},$ for $1\leq k\leq m,$ then equation (3) can be transformed
into a system of the form (1), whose associated vector field (2), written in
terms of original variables $(x,u^{(m-1)}),$ becomes

$Z=\partial_{x}+u_{1}\partial_{u}+\ldots+\phi(x,u^{(m-1)})\partial_{u_{m-1}}.$
(4)

In this case, any integral manifold of the distribution generated by the
vector field (4) corresponds to the $(m-1)$th-prolongation of a solution of
the equation (3) [1, 3, 4].

Therefore, the method of solvable structures can be applied to integrate the
given ODE (or the system of ODEs) by quadratures alone. This outcome extends
the classical result which states that an $n$th order system of $m$
differential equations, with a $mn$th-dimensional solvable algebra of Lie
point symmetries, can be integrated by quadratures. We refer the reader to [6,
Proposition 6] and [7, Section V] for further details on the application of
solvable structures to the integration of differential equations.

##  3 $\mathcal{C}^{\infty}$-structures and integrability of distributions

This notion of $\mathcal{C}^{\infty}$-symmetry for a distribution was
introduced in [42, Definition 3.2.], as a generalization of the idea of
$\mathcal{C}^{\infty}$-symmetry for ODEs [9]:

######  Definition 3.1.

A $\mathcal{C}^{\infty}$-symmetry of an involutive distribution
$\mathcal{Z}=\mathcal{S}(\\{Z_{1},\ldots,Z_{r}\\})$ is a vector field $X$ such
that the set $\\{Z_{1},\ldots,Z_{r},X\\}$ is pointwise linearly independent on
$U$ and the distribution $\mathcal{S}(\\{Z_{1},\ldots,Z_{r},X\\})$ is
involutive.

Note that by Definition 2.1 every symmetry $X$ of an involutive distribution
$\mathcal{Z}$ is also a $\mathcal{C}^{\infty}$-symmetry of $\mathcal{Z}$.

The previous notion of $\mathcal{C}^{\infty}$-symmetry of a distribution was
used in [42] to extend the concept of solvable structure as follows:

######  Definition 3.2.

[42, Definition 3.3.] Let $\mathcal{Z}$ be an involutive distribution on $U$.
An ordered set of vector fields $\langle X_{1},\ldots,X_{n-r}\rangle$ is a
$\mathcal{C}^{\infty}$-structure for $\mathcal{Z}$ if $X_{1}$ is a
$\mathcal{C}^{\infty}$-symmetry of $\mathcal{Z}$ and, for $i=2,\ldots,n-r,$
$X_{i}$ is a $\mathcal{C}^{\infty}$-symmetry of the distribution
$\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{i-1}\\}).$

Observe that a solvable structure for $\mathcal{Z}$ is a particular case of a
$\mathcal{C}^{\infty}$-structure for $\mathcal{Z}$ where each $X_{i}$ a
symmetry of $\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{i-1}\\})$ instead
of a $\mathcal{C}^{\infty}$-symmetry.

The main result concerning $\mathcal{C}^{\infty}$-structures is that they can
be used to integrate the distribution $\mathcal{Z}$ by solving successively
$n-r$ completely integrable Pfaffian equations. Unlike solvable structures,
such Pfaffian equations may not be integrable by quadratures:

######  Theorem 3.1.

[42, Theorem 3.5] Let $\mathcal{Z}$ be an involutive distribution 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.

The next subsection outlines a procedure that can be employed to integrate the
distribution $\mathcal{Z}$ when we have a $\mathcal{C}^{\infty}$-structure of
vector fields. This procedure will be used in subsequent sections to integrate
various distributions that emerge in problems modelled by differential
equations.

###  3.1 $\mathcal{C}^{\infty}$-structure-based method of integration

Given a $\mathcal{C}^{\infty}$-structure of vector fields $\langle
X_{1},\dots,X_{r}\rangle$ for $\mathcal{Z},$ a method that can be used to
integrate $\mathcal{Z}$ by applying Theorem 3.1 proceeds as follows. Let
$(x_{1},\dots,x_{n})$ be a system of local coordinates on
$U\subset\mathbb{R}^{n}$ and consider the volume form
$\boldsymbol{\Omega}=dx_{1}\wedge\cdots\wedge dx_{n}$.

Let us introduce the 1-forms

$\omega_{i}={X_{n-r}\,\lrcorner\,\ldots\,\lrcorner\,\widehat{X_{i}}\,\lrcorner\,\ldots\,\lrcorner\,X_{1}\,\lrcorner\,Z_{r}\,\lrcorner\,\ldots\,\lrcorner\,Z_{1}\,\lrcorner\,\boldsymbol{\Omega}},\quad
1\leq i\leq n-r,$ (5)

where $\widehat{X_{i}}$ indicates omission of $X_{i},$ and set

$\mathcal{P}_{i}:=\mathcal{S}(\\{\omega_{i+1},\ldots,\omega_{n-r}\\}),\quad
0\leq i\leq n-r-1.$ (6)

It follows from (5) that

$\begin{array}[]{l}\mathcal{P}_{0}=\mathcal{Z}^{\circ},\\\
\mathcal{P}_{i}=\left(\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{i}\\})\right)^{\circ},\quad
1\leq i\leq n-r-1.\end{array}$ (7)

Considering that the distribution $\mathcal{Z}$ is involutive and that,
according to Definition 3.2, the distributions
$\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{i}\\})$ for $1\leq i\leq
n-r-1$ are also involutive, then it can be deduced from (7) that the Pfaffian
systems given in (6) are completely integrable. More explicitly, there exist
1-forms $\sigma_{i}$ for $1\leq i\leq n-r$ such that

$d\omega_{i}=\sigma_{i}\wedge\omega_{i}+\sum_{j=i+1}^{n-r}\sigma_{i}^{j}\wedge\omega_{j},$
(8)

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

Since the integration of the involutive distribution $\mathcal{Z}$ is
equivalent to the integration of the Pfaffian system $\mathcal{P}_{0},$ we
describe below how to integrate $\mathcal{P}_{0}$ step by step:

  1. 1.

For $i=n-r,$ equation (8) becomes
$d\omega_{n-r}=\sigma_{n-r}\wedge\omega_{n-r},$ which implies that the
Pfaffian equation $\omega_{n-r}\equiv 0$ is Frobenius integrable. A first
integral $I_{n-r}=I_{n-r}(x_{1},\ldots,x_{n})$ for $\mathcal{P}_{n-r-1}$ is
any particular solution to the system of linear first-order PDEs arising from
the condition

$dI_{n-r}\wedge\omega_{n-r}=0.$

For $C_{n-r}\in\mathbb{R},$ the level set of $I_{n-r}$

$\Sigma_{(C_{n-r})}=\\{x\in U\subset\mathbb{R}^{n}:I_{n-r}(x)=C_{n-r}\\}$ (9)

defines an integral submanifold, of dimension $n-1,$ of the distribution
$\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{n-r-1}\\}).$

  2. 2.

For $1\leq i\leq n-r,$ we denote by $\omega_{i}|_{\Sigma_{(C_{n-r})}}$ the
restriction of $\omega_{i}$ to $\Sigma_{(C_{n-r})}.$ Observe that
$\omega_{n-r}|_{\Sigma_{(C_{n-r})}}=0.$ The restriction to
$\Sigma_{(C_{n-r})}$ of equations (8) for $1\leq i\leq i=n-r-1,$ implies that
$\omega_{n-r-1}|_{\Sigma_{(C_{n-r})}}$ is Frobenius integrable. As before, a
corresponding first integral $I_{n-r-1}=I_{n-r-1}(x;C_{n-r}),$ defined for $x$
in some open set of $\Sigma_{(C_{n-r})},$ is given by any particular solution
to the system of linear homogeneous first-order PDEs arising from the
condition

$dI_{n-r-1}\wedge\omega_{n-r-1}|_{\Sigma_{(C_{n-r})}}=0.$

For $C_{n-r-1}\in\mathbb{R},$ the submanifold of $\Sigma_{(C_{n-r})}$ defined
by the level set $I_{n-r-1}=C_{n-r-1}$ is an integral manifold of the Pfaffian
equation $\omega_{n-r-1}|_{\Sigma_{(C_{n-r})}}\equiv 0$, that will be denoted
by $\Sigma_{(C_{n-r-1},C_{n-r})}.$

  3. 3.

We continue this process, taking into account that in each stage we integrate
a 1-form defined in a space whose dimension is one unit lower than in the
previous step. At the end, we obtain the integral manifold
$\Sigma_{(C_{1},\ldots,C_{n-r})}$ of $\mathcal{Z}^{\circ},$ expressed in
implicit form as $I_{1}=C_{1},$ where $I_{1}$ denotes the first integral that
arises after integrating the last Pfaffian equation
$\omega_{1}|_{\Sigma_{(C_{2},\ldots,C_{n-r})}}\equiv 0.$

The theoretical foundation behind the procedure above is explained in [42,
Theorem 3.5]. Readers interested in a closer exploration of the
$\mathcal{C}^{\infty}$-structure integration process and in reviewing
illustrative examples are referred to Sections 3.3 and 3.4 in [43].

In addition, if an element $X_{i}$ of the $\mathcal{C}^{\infty}$-structure is
not only a $\mathcal{C}^{\infty}$-symmetry of
$\mathcal{Z}\oplus\mathcal{S}(\\{X_{1},\ldots,X_{i-1}\\})$ but a symmetry,
then the corresponding Pfaffian equation at the $i$th stage can be solved by
quadrature using a (relative) integrating factor (see Theorem 4.1 and Remark
4.3 in [43] for details). The integrability of the distribution by quadrature
via solvable structures turns out to be a special case of the more general
$\mathcal{C}^{\infty}$-structure integration method.

In the following sections we use the integration method described above to
find exact solutions to several problems modelled by ordinary differential
equations.

##  4 $\mathcal{C}^{\infty}$-structures for systems of first-order ODEs

In this section we examine the application of the
$\mathcal{C}^{\infty}$-structure method to systems of first-order ODEs.

The first system describes a Lotka-Volterra model previously considered by P.
Basarab-Horwarth in his paper on solvable structures [5]. His procedure
requires three vector fields to produce two independent first integrals of the
system. In the following subsection we show that only one of these vector
fields is needed to construct a $\mathcal{C}^{\infty}$-structure which can be
used to completely solve the system.

###  4.1 A Lotka-Volterra model

Lotka-Volterra models, also known as predator-prey models, are systems of
first order ODEs used to describe the dynamics between two or more interacting
species in an ecosystem, typically a predator and its prey. The Lotka-Volterra
model is a simple but powerful tool for understanding the dynamics of
predator-prey interactions and has applications in fields such as ecology,
biology, and economics (see, for example, [51, 52, 53, 54] for further
details).

P. Basarab-Horwath in [5, Section 4] applied a method based on solvable
structures to find two first integrals for a biparametric family of 3D Lotka-
Volterra models

$\left\\{\begin{array}[]{l}\dot{x}(t)=zx-\dfrac{yx}{AB},\\\\[7.02625pt]
\dot{y}(t)=xy+Azy,\\\\[7.02625pt] \dot{z}(t)=Bxz+yz,\\\ \end{array}\right.$
(10)

with arbitrary constants $A,B\in\mathbb{R}$, $A,B\neq 0$. More specifically,
he provided two vector fields

$\begin{array}[]{lll}Y_{1}&=&z\partial_{x}+Ay\partial_{y}+(y+Bz)\partial_{z},\\\
Y_{2}&=&x\partial_{x}-ABz\partial_{z}.\end{array}$ (11)

which are in involution with the vector field corresponding to the system:

$Z=\left(zx-\dfrac{yx}{AB}\right)\partial_{x}+\left(xy+Azy\right)\partial_{y}+\left(Bxz+yz\right)\partial_{z},$

as it can be checked through the corresponding commutation relationships.
However, neither $\langle Y_{1},Y_{2}\rangle$ nor $\langle Y_{2},Y_{1}\rangle$
constitutes a solvable structure for $\mathcal{S}(\\{Z\\}),$ because
$[Y_{1},Y_{2}]\notin\mathcal{S}(\\{Z,Y_{1}\\})$ and
$[Y_{1},Y_{2}]\notin\mathcal{S}(\\{Z,Y_{2}\\}).$ For this reason, P. Basarab-
Horwath had to provide an additional vector field

$V=x\partial_{x}+y\partial_{y}+z\partial_{z},$ (12)

which is a symmetry of $\mathcal{S}({Z})$ and commutes with $Y_{1}$ and
$Y_{2}.$ This implies that $V$ is a symmetry of both involutive distributions
$\mathcal{S}(\\{Z,Y_{1}\\})$ and $\mathcal{S}(\\{Z,Y_{2}\\})$. Applying the
theoretical results on solvable structures, the symmetry $V$ was used in [5]
to integrate, separately and by quadratures, the distributions
$\mathcal{S}(\\{Z,Y_{1}\\})$ and $\mathcal{S}(\\{Z,Y_{2}\\})$.

A first integral for $\mathcal{S}(\\{Z,Y_{1}\\})$ is

$\varphi_{1}=ABx+y-Az,$ (13)

while

$\varphi_{2}=x^{AB}y^{-B}z$ (14)

is a first integral for $\mathcal{S}(\\{Z,Y_{2}\\})$. These first integrals
are functionally independent, because $Z\wedge Y_{1}\wedge Y_{2}\neq 0.$

It is interesting to note that only one of the vector fields $Y_{1}$ or
$Y_{2}$ is necessary to integrate system (10) by the
$\mathcal{C}^{\infty}$-structure method: since $\mathcal{S}(\\{Z,Y_{1}\\})$ is
an involutive distribution, $Y_{1}$ can be chosen as the first element of a
$\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{Z\\})$. The last element
can be any vector field independent with $\\{Z,Y_{1}\\},$ such as
$\partial_{z}.$ Therefore, $\langle Y_{1},\partial_{z}\rangle$ defines a
$\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{Z\\})$ and it can be used
to integrate the system by the procedure described in Section 3.1. The same
procedure could be followed by using $Y_{2}$ instead $Y_{1},$ because $\langle
Y_{2},\partial_{z}\rangle$ is also a $\mathcal{C}^{\infty}$-structure for
$\mathcal{S}(\\{Z\\})$.

Nevertheless, instead of using one of these two
$\mathcal{C}^{\infty}$-structures, which require the knowledge of at least one
of the vector fields $Y_{1}$ or $Y_{2}$, we show how to construct a
$\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{Z\\})$ directly, without
using the vector fields provided by Basarab-Horwath. It is worth noting that
the method used to obtain these vector fields has not been explained in [5].

In order to find a $\mathcal{C}^{\infty}$-structure for
$\mathcal{S}(\\{Z\\}),$ we first observe that a if vector field $X_{1}$ is a
$\mathcal{C}^{\infty}$-symmetry of $\mathcal{S}(\\{Z\\}),$ then so is any
vector field in $\mathcal{S}(\\{Z,X_{1}\\}).$ This allows us to simplify the
search for $X_{1}$ by assuming that its form is

$X_{1}=\partial_{y}+g(x,y,z)\partial_{z}.$

According to Definition 3.2, $X_{1}$ must satisfy the condition
$[X_{1},Z]\in\mathcal{S}(\\{Z,X_{1}\\}).$ Equivalently, the 1-form
$\omega_{2}=X_{1}\,\lrcorner\,Z\,\lrcorner\,\boldsymbol{\Omega},$ where
$\boldsymbol{\Omega}=dx\wedge dy\wedge dz,$ satisfies $\omega_{2}\wedge
d\omega_{2}=0,$ i.e., the Pfaffian equation $\omega_{2}\equiv 0$ is completely
integrable. Any of these two equivalent conditions yields a determining
equation for the function $g=g(x,y,z).$ It can be checked that such PDE is of
the form

$\rho_{1}\,g_{x}+\rho_{2}\,g_{y}+\rho_{3}\,g_{z}+(Ag-1)(\rho_{4}\,g+\rho_{5})=0,$
(15)

where we omit the explicit expressions of the functions
$\rho_{i}=\rho_{i}(x,y,z),$ for $1\leq i\leq 5,$ because they are irrelevant
for the following discussion. A particular solution of the determining
equation (15) arises immediately, the constant function

$g(x,y,z)=\dfrac{1}{A}.$ (16)

Therefore, the vector field

$X_{1}=\partial_{y}+\frac{1}{A}\partial_{z}$ (17)

is a $\mathcal{C}^{\infty}$-symmetry of $\mathcal{S}(\\{Z\\})$. As the second
element of the $\mathcal{C}^{\infty}$-structure we can choose any vector field
$X_{2}$ such that $\\{Z,X_{1},X_{2}\\}$ are linearly independent. For
instance, we can use the vector field $X_{2}=\partial_{z}$.

Once the $\mathcal{C}^{\infty}$-structure $\langle X_{1},X_{2}\rangle$ for
$\mathcal{S}(\\{Z\\})$ has been determined, we calculate the 1-forms
$\omega_{1}$ and $\omega_{2}$ given in (5):

$\begin{array}[]{rl}\omega_{1}=X_{2}\,\lrcorner\,Z\,\lrcorner\,\boldsymbol{\Omega}&=\left(Ayz+xy\right)dx-\dfrac{x\left(ABz-y\right)}{AB}dy,\\\\[10.0pt]
\omega_{2}=X_{1}\,\lrcorner\,Z\,\lrcorner\,\boldsymbol{\Omega}&=-\dfrac{x\left(ABz-y\right)}{A^{2}B}\left(ABdx+dy-
Adz\right).\end{array}$ (18)

The Pfaffian equation $\omega_{2}\equiv 0$ is completely integrable and a
corresponding first integral $I_{2}=I_{2}(x,y,z)$ arises from the condition
$dI_{2}\wedge\omega_{2}=0,$ which yields the following system of PDEs:

$\begin{array}[]{lll}A(I_{2})_{y}+(I_{2})_{z}&=&0,\\\
(I_{2})_{x}+B(I_{2})_{z}&=&0.\end{array}$ (19)

The first equation implies that $I_{2}=F\left(x,r\right),$ where $r=Az-y$ and
$F=F(x,r)$ is, in principle, an arbitrary smooth function. Then the second
equation in (19) becomes

$F_{x}+ABF_{r}=0,$

from which the particular solution $F(x,r)=ABx-r$ arises immediately.
Therefore, a first integral for $\omega_{2}\equiv 0$ is given by
$I_{2}=F\left(x,Az-y\right):$

$I_{2}=ABx-Az+y.$ (20)

Observe that $I_{2}=\varphi_{1},$ where $\varphi_{1}$ is the first integral
(13) provided by Basarab-Horwath. In order to find the remaining first
integral, we restrict $\omega_{1}$ to the submanifold $\Sigma_{(C_{2})}$
implicitly defined by $I_{2}=C_{2},$ where $C_{2}\in\mathbb{R}:$

$\omega_{1}|_{\Sigma_{(C_{2})}}=\left(ABxy-
C_{2}y+xy+y^{2}\right)dx-\frac{x\left(AB^{2}x-BC_{2}+By-y\right)}{AB}dy.$

The Pfaffian equation $\omega_{1}|_{\Sigma_{(C_{2})}}\equiv 0$ is completely
integrable. It can be checked that

$\mu=\dfrac{1}{xy(ABx+y-C_{2})}$

is an integrating factor for $\omega_{1}|_{\Sigma_{(C_{2})}}.$ A corresponding
primitive $I_{1}=I_{1}(x,y;C_{2})$ arises after integrating two rational
functions:

${I}_{1}=\ln(|x^{AB}y^{-B}(ABx-C_{2}+y)|).$ (21)

If $C_{2}$ in (21) is replaced by the right-hand side of (20) we get the
function $J_{1}(x,y,z)=I_{1}(x,y;I_{2}):$

$J_{1}=Ax^{AB}y^{-B}z,$

which, up to a constant, coincides with the first integral $\varphi_{2}$ in
(14), previously obtained in [5].

The orbits of the system (10) can be expressed in implicit form as follows:

$Ax^{AB}y^{-B}z=C_{1},\quad ABx-Az+y=C_{2},\quad(C_{1},C_{2}\in\mathbb{R}).$
(22)

In consequence, the $\mathcal{C}^{\infty}$-structure method provides an
alternative approach to integrate system (10). In this procedure, only the
vector field (17) has been used, instead of the three vector fields
$Y_{1},Y_{2}$ and $V$ in (11) and (12) required in [5] by using solvable
structures techniques.

###  4.2 Integration of a non-autonomous system through
$\mathcal{C}^{\infty}$-structures

In the following example we study a system of first order ODEs which, to our
knowledge, can not be easily solved by classical procedures. We will show how
to construct a $\mathcal{C}^{\infty}$-structure for the system and how to use
it to find its general solution, expressed in terms of a complete set of
solutions of a linear second-order homogeneous equation.

Consider the system of first-order ODEs:

$\left\\{\begin{array}[]{l}\dot{x}(t)=\dfrac{2ty-t^{2}x^{2}-2yx^{2}-x^{2}}{2tx},\\\\[7.02625pt]
\dot{y}(t)=t-x^{2}y,\end{array}\right.$ (23)

with associated vector field

$Z=\partial_{t}+\dfrac{2ty-t^{2}x^{2}-2yx^{2}-x^{2}}{2tx}\partial_{x}+(t-x^{2}y)\partial_{y},$

defined on the open set

$M=\\{(t,x,y)\in\mathbb{R}^{3}:tx\neq 0\\}.$ (24)

To find the first element $X_{1}$ of a $\mathcal{C}^{\infty}$-structure for
the distribution $\mathcal{S}(\\{Z\\})$ we assume, as in the previous example,
that $X_{1}$ is of the form $X_{1}=\partial_{x}+g(t,x,y)\partial_{y}.$ The
determining equation for the function $g(t,x,y)$ can be obtained from the
condition $[X_{1},Z]\in\mathcal{S}(\\{Z,X_{1}\\}).$ This is equivalent to the
condition $\omega_{2}\wedge d\omega_{2}=0,$ where
$\omega_{2}=X_{1}\,\lrcorner\,Z\,\lrcorner\,\boldsymbol{\Omega}$ for
$\boldsymbol{\Omega}=dt\wedge dx\wedge dy.$

In order to ease the search of a particular solution of this determining
equation, we can begin by trying to find a particular solution of the form
$g(t,x,y)=f(t)h(x).$ It can be checked that by canceling out the coefficients
of $y$ we obtain a system of determining equations for the functions $f=f(t)$
and $h=h(x)$ that, after some calculations, becomes

$h(x)f(t)=tx,\quad f^{\prime}(t)=\dfrac{f(t)}{t}.$

By choosing the particular solution

$h(x)=x,\quad f(t)=t,$

we obtain that the vector field $X_{1}=\partial_{x}+tx\partial_{y}$ is a
$\mathcal{C}^{\infty}$-symmetry of the distribution $\mathcal{S}(\\{Z\\})$ and
hence it can be selected as the first element of a
$\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{Z\\}).$ As a second
element we can choose any vector field $X_{2}$ such that the set
$\\{Z,X_{1},X_{2}\\}$ is linearly independent, such as, for instance,
$X_{2}=\partial_{y}.$ Therefore the vector fields

$X_{1}=\partial_{x}+tx\partial_{y},\quad X_{2}=\partial_{y}$ (25)

constitute a $\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{Z\\}).$ The
corresponding commutations relationships become

$\displaystyle[X_{1},Z]=\frac{-2tx^{4}+t^{2}x^{2}-2x^{2}y-2ty-x^{2}}{tx^{2}}\,X_{1},$
(26) $\displaystyle[X_{2},Z]=\frac{-x^{2}+t}{tx}\,X_{1}-t\,X_{2},$ (27)
$\displaystyle[X_{2},X_{1}]=0.$ (28)

It is crucial to emphasize that neither $X_{1}$ is a symmetry of
$\mathcal{S}({Z})$, nor $X_{2}$ is a symmetry of $\mathcal{S}({Z,X_{1}})$.
Specifically, $X_{1}$ and $X_{2}$ do not correspond to symmetries of the
system (23). As a result, the integration method based on the
$\mathcal{C}^{\infty}$-structure presented here provides a novel alternative
to conventional symmetry procedures.

The integration procedure by using the $\mathcal{C}^{\infty}$-structure
defined by (25) proceeds as follows: the corresponding 1-forms given in (5)
become

$\displaystyle\omega_{1}=\dfrac{2ty-t^{2}x^{2}-2x^{2}y-x^{2}}{2xt}dt-dx,$ (29)
$\displaystyle\omega_{2}=\left(ty-\frac{1}{2}t^{2}x^{2}-\frac{1}{2}x^{2}-t\right)dt-
txdx+dy.$ (30)

The Pfaffian equation $\omega_{2}\equiv 0$ is completely integrable; it can be
checked that a corresponding first integral is given by the smooth function

$I_{2}(t,x,y)=\frac{1}{2}e^{\frac{1}{2}t^{2}}(tx^{2}-2y+2).$ (31)

The restriction of the 1-form $\omega_{1}$ given in (29) to the level set
$I_{2}=C_{2}$, $C_{2}\in\mathbb{R},$ denoted by ${\Sigma_{(C_{2})}},$ becomes

$\omega_{1}|_{\Sigma_{(C_{2})}}=\dfrac{2t-tx^{4}-3x^{2}+2C_{2}(t-x^{2})e^{-\frac{1}{2}t^{2}}}{2tx}dt-
dx.$ (32)

In order to solve the Pfaffian equation $\omega_{1}|_{\Sigma_{(C_{2})}}\equiv
0,$ we introduce the change $\bar{x}=x^{2}$ which transforms the ODE
associated to the Pfaffian equation into the Riccati-type equation

$\bar{x}^{\prime}(t)=-\bar{x}(t)^{2}-\dfrac{2C_{2}e^{-\frac{1}{2}t^{2}}+3}{t}\bar{x}(t)+2(C_{2}e^{-\frac{1}{2}t^{2}}+1).$
(33)

The standard transformation $\bar{x}(t)=\psi^{\prime}(t)/\psi(t)$ transforms
the Riccati-type equation (33) into the following linear second-order
homogeneous ODE:

$\psi^{\prime\prime}(t)+\left(\dfrac{2C_{2}e^{-\frac{1}{2}t^{2}}+3}{t}\right)\psi^{\prime}(t)-2(C_{2}e^{-\frac{1}{2}t^{2}}+1)\psi(t)=0.$
(34)

Let $\psi_{1}=\psi_{1}(t;C_{2})$ and $\psi_{2}=\psi_{2}(t;C_{2})$ be a
fundamental set of solutions to the linear ODE (34). These functions can be
used to express a first integral associated to the Riccati equation (33) (see,
for instance, Proposition 4.1 in [55]). As a consequence, a first integral of
the Pfaffian equation defined by (32) becomes:

$I_{1}(t,x;C_{2})=\dfrac{-x^{2}\psi_{1}(t;C_{2})+\psi_{1}^{\prime}(t;C_{2})}{-x^{2}\psi_{2}(t;C_{2})+\psi_{2}^{\prime}(t;C_{2})}.$
(35)

By replacing $C_{2}$ by the right-hand side of (31) we get the function
$J_{1}(t,x,y)=I_{1}(t,x;I_{2})$, which is a first integral of
$\mathcal{S}(\\{Z\\})$:

$J_{1}(t,x,y)=\dfrac{-x^{2}\psi_{1}(t;\frac{1}{2}e^{\frac{1}{2}t^{2}}(tx^{2}-2y+2))+\psi_{1}^{\prime}(t;\frac{1}{2}e^{\frac{1}{2}t^{2}}(tx^{2}-2y+2))}{-x^{2}\psi_{2}(t;\frac{1}{2}e^{\frac{1}{2}t^{2}}(tx^{2}-2y+2))+\psi_{2}^{\prime}(t;\frac{1}{2}e^{\frac{1}{2}t^{2}}(tx^{2}-2y+2))}.$
(36)

From $I_{1}(t,x;C_{2})=C_{1}$ and $I_{2}(t,x,y)=C_{2}$ where
$C_{1},C_{2}\in\mathbb{R},$ we get the general solution to system (23):

$\left\\{\begin{array}[]{lll}x(t)&=&\pm\left(\dfrac{C_{1}\psi_{2}^{\prime}(t;C_{2})-\psi_{1}^{\prime}(t;C_{2})}{C_{1}\psi_{2}(t;C_{2})-\psi_{1}(t;C_{2})}\right)^{1/2},\\\
y(t)&=&1+C_{2}e^{-\frac{1}{2}t^{2}}+\dfrac{t}{2}\,\dfrac{C_{1}\psi_{2}^{\prime}(t;C_{2})-\psi_{1}^{\prime}(t;C_{2})}{C_{1}\psi_{2}(t;C_{2})-\psi_{1}(t;C_{2})}.\end{array}\right.$
(37)

where $\psi_{1}=\psi_{1}(t;C_{2})$ and $\psi_{2}=\psi_{2}(t;C_{2})$ are two
functionally independent solutions to the linear ODE (34).

####  4.2.1 Some particular families of solutions

For particular values of the arbitrary constant $C_{2}\in\mathbb{R},$ the
solutions to the corresponding linear ODE (34) are well-known special
functions. For instance, for $C_{2}=0,$ equation (34) becomes

$t^{2}\psi^{\prime\prime}(t)+3t\psi^{\prime}(t)-2t^{2}\psi(t)=0.$ (38)

Through the change of variables

$z=\sqrt{2}t,\quad\phi(z)=t\psi(t),$ (39)

equation (38) becomes the modified Bessel equation

$z^{2}\phi^{\prime\prime}(z)+z\phi^{\prime}(z)-(1+z^{2})\phi(z)=0.$ (40)

A fundamental set of solutions to equation (40) are the modified Bessel
functions $\mathbf{I}_{1}$ and $\mathbf{K}_{1}$ of the first and second kinds,
respectively [56]. Therefore, according to (39), the functions

$\psi_{1}(t)=\dfrac{1}{t}\mathbf{I}_{1}(\sqrt{2}t),\qquad\psi_{2}(t)=\dfrac{1}{t}\mathbf{K}_{1}(\sqrt{2}t),$
(41)

are two linearly independent solutions to equation (38). As a consequence, a
1-parameter family of solutions to system (23), which corresponds to (37) when
$C_{2}=0,$ can be expressed in terms of the modified Bessel functions as
follows:

$\left\\{\begin{array}[]{lll}x(t)&=&\pm\left(\sqrt{2}\dfrac{C_{1}\mathbf{K}_{1}^{\prime}(\sqrt{2}t)-\mathbf{I}_{1}^{\prime}(\sqrt{2}t)}{C_{1}\mathbf{K}_{1}(\sqrt{2}t)-\mathbf{I}_{1}(\sqrt{2}t)}-\dfrac{1}{t}\right)^{1/2},\\\
y(t)&=&\dfrac{1}{2}+\dfrac{\sqrt{2}t}{2}\,\dfrac{C_{1}\mathbf{K}_{1}^{\prime}(\sqrt{2}t)-\mathbf{I}_{1}^{\prime}(\sqrt{2}t)}{C_{1}\mathbf{K}_{1}(\sqrt{2}t)-\mathbf{I}_{1}(\sqrt{2}t)}.\end{array}\right.$
(42)

The derivatives of the modified Bessel functions $\mathbf{I}_{1}$ and
$\mathbf{K}_{1}$ can be expressed in terms of the modified Bessel functions
$\mathbf{I}_{0}$ and $\mathbf{K}_{0}$ [56]:

$\mathbf{K}_{1}^{\prime}(z)=\mathbf{K}_{0}(z)-\dfrac{1}{z}\mathbf{K}_{1}(z),\quad\mathbf{I}_{1}^{\prime}(z)=\mathbf{I}_{0}(z)-\dfrac{1}{z}\mathbf{I}_{1}(z).$

Then

$\mathbf{K}_{1}^{\prime}(\sqrt{2}t)=-\mathbf{K}_{0}(\sqrt{2}t)-\dfrac{\sqrt{2}}{2t}\mathbf{K}_{1}(\sqrt{2}t),\quad\mathbf{I}_{1}^{\prime}(\sqrt{2}t)=\mathbf{I}_{0}(\sqrt{2}t)-\dfrac{\sqrt{2}}{2t}\mathbf{I}_{1}(\sqrt{2}t),$

and therefore (42) becomes

$\left\\{\begin{array}[]{lll}x(t)&=&\pm\left(\sqrt{2}\dfrac{C_{1}\mathbf{K}_{0}(\sqrt{2}t)-\mathbf{I}_{0}(\sqrt{2}t)}{C_{1}\mathbf{K}_{1}(\sqrt{2}t)-\mathbf{I}_{1}(\sqrt{2}t)}-\dfrac{2}{t}\right)^{1/2},\\\
y(t)&=&\dfrac{\sqrt{2}t}{2}\,\dfrac{C_{1}\mathbf{K}_{0}(\sqrt{2}t)-\mathbf{I}_{0}(\sqrt{2}t)}{C_{1}\mathbf{K}_{1}(\sqrt{2}t)-\mathbf{I}_{1}(\sqrt{2}t)}.\end{array}\right.$
(43)

##  5 $\mathcal{C}^{\infty}$-structures for scalar ODEs with a lack of Lie
point symmetries

In this section we present a collection of ordinary differential equations
whose symmetry algebras are either trivial or of lower dimension than the
order of the ODE. In the latter scenario, the Lie method encounters certain
obstacles when attempting to obtain the general solution. However, we
demonstrate how the $\mathcal{C}^{\infty}$-structures method successfully
overcomes these difficulties and provides exact solutions of the equations
under investigation.

###  5.1 A third-order ODE with two-dimensional algebra of Lie point
symmetries

In this example we consider a third-order ODE:

$u_{3}=(u_{1}-u)u_{2}-\dfrac{u_{1}^{2}}{2}+u_{1}+\dfrac{u^{2}}{2},$ (44)

whose associated vector field is

$Z=\partial_{x}+u_{1}\partial_{u}+u_{2}\partial_{u_{1}}+\left((u_{1}-u)u_{2}-\frac{1}{2}(u_{1}^{2}-u^{2})+u_{1}\right)\partial_{u_{2}}.$

It can be checked that the symmetry algebra of equation (44) is two
dimensional and spanned by $\partial_{x}$ and $e^{x}\partial_{u}.$ By
employing the Lie method of reduction, the transformation

$z=u_{1}-u,\quad h(z)=u_{2}-u_{1},$ (45)

leads to the first-order ODE

$h(z)h^{\prime}(z)=(z-1)h(z)+\dfrac{z^{2}}{2}.$ (46)

Equation (46) is an Abel-type equation whose general solution can be expressed
in implicit form in terms of the modified Bessel functions of the first and
second kinds $\mathbf{I}_{\alpha}$ and $\mathbf{K}_{\alpha},$ for $\alpha=0,1$
[56]:

$\dfrac{z\mathbf{K}_{0}\left(-\sqrt{z^{2}-{2}{h(z)}}\right)-\sqrt{z^{2}-{2}{h(z)}}\mathbf{K}_{1}\left(-\sqrt{z^{2}-{2}{h(z)}}\right)}{z\mathbf{I}_{0}\left(\sqrt{z^{2}-{2}{h(z)}}\right)-\sqrt{z^{2}-{2}{h(z)}}\mathbf{I}_{1}\left(\sqrt{z^{2}-{2}{h(z)}}\right)}=C_{1}.$
(47)

The recovery of solutions to equation (44) from (47), by means of the
transformation (45), seems to be infeasible.

For this reason, we intend to integrate equation (44) by using the
$\mathcal{C}^{\infty}$-structures method. Similar to the previous examples,
finding the elements of a $\mathcal{C}^{\infty}$-structure can significantly
be simplified by assuming some of the infinitesimals to be constant or linear
in $u_{1}$. By following this approach, we obtain the following independent
vector fields

$\begin{array}[]{l}X_{1}=\partial_{u}+\partial_{u_{1}}+\partial_{u_{2}},\\\
X_{2}=\partial_{u_{1}}+(u_{1}-u+1)\partial_{u_{2}},\\\
X_{3}=\partial_{u_{2}}.\\\ \end{array}$

They form a $\mathcal{C}^{\infty}$-structure for $\mathcal{S}(\\{Z\\}),$ as
can be verified using the Lie brackets:

$\begin{array}[]{l}[X_{1},Z]=X_{1},\\\ [X_{2},Z]=X_{1}+(u_{1}-u)X_{2},\\\
[X_{2},X_{1}]=0.\\\ \end{array}$

It is important to emphasize that neither $X_{1}$ is a symmetry of
$\mathcal{S}(\\{Z\\}),$ nor $X_{2}$ is a symmetry of
$\mathcal{S}(\\{Z,X_{1}\\}).$ In particular, nor $X_{1}$ neither $X_{2}$
correspond to symmetries of equation (44).

We use the volume form $\boldsymbol{\Omega}=dx\wedge du\wedge du_{1}\wedge
du_{2}$ to construct the corresponding 1-forms given in (5):

$\begin{array}[]{l}\omega_{1}=-u_{1}dx+du,\\\ \omega_{2}=(u_{2}-u_{1})dx+du-
du_{1},\\\
\omega_{3}=(u_{2}+uu_{1}-u_{1}+\frac{1}{2}(u_{1}^{2}+u^{2}))dx+(u_{1}-u)du+(u-u_{1}-1)du_{1}+du_{2}.\\\
\end{array}$ (48)

A first integral for the first Pfaffian equation $\mathcal{P}_{2},$ i.e., a
function $I_{3}=I_{3}(x,u,u_{1},u_{2})$ such that $dI_{3}\wedge\omega_{3}=0$,
is given by

$I_{3}=e^{x}\left(u_{2}-\frac{1}{2}(u_{1}-u)^{2}-u_{1}\right).$

Let $\Sigma_{(C_{3})}$ denote, as before, the level set $I_{3}=C_{3},$ for
$C_{3}\in\mathbb{R}.$ The restriction of the 1-form $\omega_{2}$ in (48) to
$\Sigma_{(C_{3})}$ becomes

$\omega_{2}|_{\Sigma_{(C_{3})}}=\left(\frac{1}{2}(u_{1}-u)^{2}+C_{3}e^{-x}\right)dx+du-
du_{1}.$ (49)

In order to continue the integration process, we need to distinguish the
following cases:

  1. 1.

Case I: $C_{3}>0.$

It can be checked that a function $I_{2}=I_{2}(x,u,u_{1};C_{3})$ such that
$dI_{2}\wedge\omega_{2}|_{\Sigma_{(C_{3})}}=0$ becomes:

$I_{2}=\dfrac{e^{x/2}(u_{1}-u)\mathbf{J}_{0}({\sqrt{2C_{3}}e^{-x/2}})+\sqrt{2C_{3}}\mathbf{J}_{1}(\sqrt{2C_{3}}e^{-x/2})}{e^{x/2}(u-u_{1})\mathbf{Y}_{0}(\sqrt{2C_{3}}e^{-x/2})-\sqrt{2C_{3}}\mathbf{Y}_{1}(\sqrt{2C_{3}}e^{-x/2})},$

where $\mathbf{J}_{\alpha},\mathbf{Y}_{\alpha}$ are the Bessel functions of
first and second kind, respectively [56].

Let $\Sigma_{(C_{2},C_{3})}$ denote the submanifold of $\Sigma_{(C_{3})}$
defined by $I_{2}(x,u,u_{1};C_{3})=C_{2},$ where $C_{2}\in\mathbb{R}.$ The
restriction of the 1-form $\omega_{3}$ in (48) to $\Sigma_{(C_{2},C_{3})}$
becomes

$\omega_{1}|_{\Sigma_{(C_{2},C_{3})}}=\left(\dfrac{{\sqrt{2C_{3}}e^{-x/2}}\left(C_{2}\mathbf{Y}_{1}(\sqrt{2C_{3}}e^{-x/2})+\mathbf{J}_{1}(\sqrt{2C_{3}}e^{-x/2})\right)}{C_{2}\mathbf{Y}_{0}(\sqrt{2C_{3}}e^{-x/2})+\mathbf{J}_{0}(\sqrt{2C_{3}}e^{-x/2})}-u\right)dx+du.$

A function $I_{1}=I_{1}(x,u;C_{2},C_{3})$ such that
$dI_{1}\wedge\omega_{1}|_{\Sigma_{(C_{2},C_{3})}}=0$ is given by

$I_{1}=\sqrt{2C_{3}}\psi_{(C_{2};C_{3})}(x)+e^{-x}u.$

where

$\psi_{(C_{2};C_{3})}^{\prime}(x)=\dfrac{C_{2}\mathbf{Y}_{1}(\sqrt{2C_{3}}e^{-x/2})+\mathbf{J}_{1}(\sqrt{2C_{3}}e^{-x/2})}{C_{2}\mathbf{Y}_{0}(\sqrt{2C_{3}}e^{-x/2})+\mathbf{J}_{0}(\sqrt{2C_{3}}e^{-x/2})}e^{-\frac{3}{2}x}.$
(50)

Finally, the solution of (44) is obtained by setting
$I_{1}(x,u;C_{2},C_{3})=C_{1},$ for $C_{1}\in\mathbb{R},$ which gives

$u(x)=-\sqrt{2C_{3}}e^{x}\psi_{(C_{2};C_{3})}(x)+C_{1}e^{x},$

where the function $\psi_{(C_{2};C_{3})}$ satisfies (50).

  2. 2.

Case II: $C_{3}<0.$

In this case a function $I_{2}=I_{2}(x,u,u_{1};C_{3})$ such that
$dI_{2}\wedge\omega_{2}|_{\Sigma_{(C_{3})}}=0$ is given by:

$I_{2}=\dfrac{e^{x/2}(u_{1}-u)\mathbf{I}_{0}\left(\sqrt{-2C_{3}}e^{-x/2}\right)-\sqrt{-2C_{3}}\mathbf{I}_{1}\left(\sqrt{-2C_{3}}e^{-x/2}\right)}{e^{x/2}(u-u_{1})\mathbf{K}_{0}\left(\sqrt{-2C_{3}}e^{-x/2}\right)-\sqrt{-2C_{3}}\mathbf{K}_{1}\left(\sqrt{-2C_{3}}e^{-x/2}\right)}$

where $\mathbf{I}_{\alpha},\mathbf{K}_{\alpha}$ are the modified Bessel
functions of first and second kind, respectively [56].

Proceeding as in the previous case, we obtain the following solution to
equation (44):

$u(x)=\sqrt{-2C_{3}}e^{x}\varphi_{(C2;C_{3})}(x)+C_{1}e^{x},$

where

$\varphi_{(C2;C_{3})}^{\prime}(x)=\dfrac{C_{2}\mathbf{K}_{1}(-\sqrt{-2C_{3}}e^{-x/2})-\mathbf{I}_{1}(\sqrt{-2C_{3}}e^{-x/2})}{C_{2}\mathbf{K}_{0}(-\sqrt{-2C_{3}}e^{-x/2})+\mathbf{I}_{0}(\sqrt{-2C_{3}}e^{-x/2})}e^{-\frac{3}{2}x}.$

  3. 3.

Case III: $C_{3}=0.$

It can be checked that a solution for the Pfaffian equation defined by the
restriction of the 1-form $\omega_{2}$ in (48) to the level set $I_{3}=0$ is
given by

$I_{2}=\frac{x}{2}+\frac{1}{u_{1}-u}.$

The restriction of the 1-form $\omega_{1}$ in (48) to the submanifold
$\Sigma_{(C_{2},0)}$ implicitly defined by $I_{3}=0,I_{2}=C_{2},$
$C_{2}\in\mathbb{R}$ becomes

$\omega_{1}|_{\Sigma_{(C_{2},0)}}=-\left(u+\frac{2}{2C_{2}-x}\right)dx+du.$

The solution of the Pfaffian equation $\omega_{1}|_{\Sigma_{(C_{2},0)}}\equiv
0$ is defined by the function

$I_{1}=ue^{-x}-2e^{-2C_{2}}\mbox{E}_{1}(x-2C_{2}),$

where $\mbox{E}_{1}=\mbox{E}_{1}(z)$ denotes the exponential integral function
[56]

$\mbox{E}_{1}(z)=\int_{z}^{\infty}\dfrac{e^{-t}}{t}dt.$

By setting $I_{1}(x,u,C_{2})=C_{1},$ for $C_{1}\in\mathbb{R},$ we finally
obtain the following 2-parameter family of exact solutions for equation (44):

$u(x)=2e^{x-2C_{2}}\mbox{E}_{1}(x-2C_{2})+C_{1}e^{x}.$ (51)

###  5.2 A fourth-order ODE with a 1-dimensional algebra of Lie point
symmetries

In this subsection we consider the fourth-order equation

$u_{4}=u_{1}u_{3}+(x^{2}+1)u_{2}+u_{2}^{2}-\frac{1}{2}(x^{2}+1)u_{1}^{2},$
(52)

which has only the Lie point symmetry $\mathbf{v}=\partial_{u}$. It can be
checked that the Lie reduction method leads to a third-order equation from
which it seems difficult to recover the general solution of the initial
equation (52).

By proceeding as in the previous examples, a $\mathcal{C}^{\infty}$-structure
$\langle X_{1},X_{2},X_{3},X_{4}\rangle$ for the distribution generated by the
vector field

$Z=\partial_{x}+u_{1}\partial_{u}+u_{2}\partial_{u_{1}}+u_{3}\partial_{u_{2}}+\left(u_{1}u_{3}+(x^{2}+1)u_{2}+u_{2}^{2}-\frac{1}{2}(x^{2}+1)u_{1}^{2}\right)\partial_{u_{3}}$

can be explicitly determined by the following vector fields:

$\begin{array}[]{l}X_{1}=\partial_{u},\\\
X_{2}=\partial_{u_{1}}+u_{1}\partial_{u_{2}}+(u_{1}^{2}+u_{2})\partial_{u_{3}},\\\
X_{3}=\partial_{u_{2}}+(x+u_{1})\partial_{u_{3}},\\\
X_{4}=\partial_{u_{3}}.\end{array}$ (53)

Since $X_{1}=\partial_{u}=\mathbf{v}^{(3)},$ where $\mathbf{v}^{(3)}$ denotes
the third-order prolongation of the Lie point symmetry $\mathbf{v}$ [1], it is
clear that $X_{1}$ is a $\mathcal{C}^{\infty}$-symmetry of
$\mathcal{S}(\\{Z\\})$ in the sense of Definition 3.1. The vector field
$X_{2}$ is a $\mathcal{C}^{\infty}$-symmetry of $\mathcal{S}(\\{Z,X_{1}\\})$
because

$\begin{array}[]{l}[X_{2},Z]=X_{1}+u_{1}X_{2},\\\ [X_{2},X_{1}]=0.\\\
\end{array}$

The vector field $X_{3}$ is a $\mathcal{C}^{\infty}$-symmetry of
$\mathcal{S}(\\{Z,X_{1},X_{2}\\})$, since

$\begin{array}[]{l}[X_{3},Z]=X_{2}+xX_{3},\\\ [X_{3},X_{1}]=0,\\\
[X_{3},X_{2}]=0.\\\ \end{array}$

Finally, $X_{4}$ is a $\mathcal{C}^{\infty}$-symmetry of
$\mathcal{S}(\\{Z,X_{1},X_{2},X_{3}\\})$ because $\\{Z,X_{1},X_{2},X_{3}\\}$
are pointwise linearly independent. In this example, $X_{2},X_{3}$ and $X_{4}$
do not correspond to symmetries of equation (52).

We use the volume form $\boldsymbol{\Omega}=dx\wedge du\wedge du_{1}\wedge
du_{2}\wedge du_{3}$ to calculate 1-forms given by (5):

$\begin{array}[]{l}\omega_{1}=u_{1}dx-du,\\\ \omega_{2}=-u_{2}dx+du_{1},\\\
\omega_{3}=-(u_{1}u_{2}-u_{3})dx+u_{1}du_{1}-du_{2},\\\
\omega_{4}=(-u_{2}(x^{2}+xu_{1}+1)+\frac{1}{2}(x^{2}+1)u_{1}^{2}+xu_{3})dx+(xu_{1}-u_{2})du_{1}-(x+u_{1})du_{2}+du_{3}.\end{array}$

  1. 1.

We begin by solving the Pfaffian equation $\omega_{4}\equiv 0$. It can be
checked that a smooth function $I_{4}=I_{4}(x,u,u_{1},u_{2},u_{3})$ such that
$dI_{4}\wedge\omega_{4}=0$ is given by:

$I_{4}=\left(\frac{1}{2}xu_{1}^{2}-u_{1}u_{2}-xu_{2}+u_{3}\right)e^{\frac{1}{2}x^{2}}.$

  2. 2.

The restriction of $\omega_{3}$ to the submanifold $\Sigma_{(C_{4})}$
implicitly defined by $I_{4}=C_{4},$ $C_{4}\in\mathbb{R},$ becomes

$\omega_{3}|_{\Sigma_{(C_{4})}}=\left(-\frac{1}{2}xu_{1}^{2}+xu_{2}+C_{4}e^{-\frac{1}{2}x^{2}}\right)dx+u_{1}du_{1}-du_{2}.$

A smooth function $I_{3}=I_{3}(x,u,u_{1},u_{2};C_{4})$ such that
$dI_{3}\wedge\omega_{3}|_{\Sigma_{(C_{4})}}=0$ can be expressed in the form:

$I_{3}=-\frac{1}{2}C_{4}\sqrt{\pi}\mbox{Erf}(x)+\left(u_{2}-\frac{1}{2}u_{1}^{2}\right)e^{-\frac{1}{2}x^{2}},$

where $\mbox{Erf}=\mbox{Erf}(z)$ denotes the error function defined by [56]

$\mbox{Erf}(z)=\dfrac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt.$ (54)

  3. 3.

The restriction of $\omega_{2}$ to the submanifold $\Sigma_{(C_{3},C_{4})}$ of
$\Sigma_{(C_{4})}$ implicitly defined by $I_{3}(x,u,u_{1},u_{2};C_{4})=C_{3},$
where $C_{3}\in\mathbb{R},$ becomes

$\omega_{2}|_{\Sigma_{(C_{3},C_{4})}}=-\left(\frac{1}{2}C_{4}\sqrt{\pi}\mbox{Erf}(x)e^{\frac{1}{2}x^{2}}+\frac{1}{2}u_{1}^{2}+C_{3}e^{\frac{1}{2}x^{2}}\right)dx+du_{1}.$

It can be checked that a function $I_{2}=I_{2}(x,u,u_{1};C_{3},C_{4})$ such
that $dI_{2}\wedge\omega_{2}|_{\Sigma_{(C_{3},C_{4})}}=0$ is given by

$I_{2}=-\frac{u_{1}\psi_{2}(x;C_{3},C_{4})+2\psi_{2}^{\prime}(x;C_{3},C_{4})}{u_{1}\psi_{1}(x;C_{3},C_{4})+2\psi_{1}^{\prime}(x;C_{3},C_{4})},$
(55)

where $\psi_{1}=\psi_{1}(x;C_{3},C_{4})$ and
$\psi_{1}=\psi_{1}(x;C_{3},C_{4})$ constitute a fundamental set of solutions
to the following two-parameter family of Schrödinger-type equations:

$\psi^{\prime\prime}(x)=-\frac{1}{2}e^{\frac{1}{2}x^{2}}\left(\sqrt{\pi}\frac{C_{4}}{2}\mbox{Erf}(x)+C_{3}\right)\psi(x).$
(56)

  4. 4.

Finally, the restriction of $\omega_{1}$ to the submanifold
$\Sigma_{(C_{2},C_{3},C_{4})}$ of $\Sigma_{(C_{3},C_{4})}$ defined by
$I_{2}(x,u,u_{1};C_{3},C_{4})=C_{2}$, with $C_{2}\in\mathbb{R}$, becomes

$\omega_{1}|_{\Sigma_{(C_{2},C_{3},C_{4})}}=\frac{2\left(C_{2}\psi_{1}^{\prime}(x;C_{3},C_{4})+\psi_{2}^{\prime}(x;C_{3},C_{4})\right)}{C_{2}\psi_{1}(x;C_{3},C_{4})+\psi_{2}(x;C_{3},C_{4})}dx+du.$

A function $I_{1}=I_{1}(x,u;C_{2},C_{3},C_{4})$ such that
$dI_{1}\wedge\omega_{1}|_{\Sigma_{(C_{2},C_{3},C_{4})}}=0$ can be calculated
by a simple quadrature and becomes

$I_{1}=u+2\ln\left(C_{2}\psi_{1}(x;C_{3},C_{4})+\psi_{2}(x;C_{3},C_{4})\right).$

As a result of the previous procedure of integration, by using the
$\mathcal{C}^{\infty}$-structure defined by (53), the initial fourth-order
equation (52) has been completely integrated. Its general solution can be
expressed in terms of a fundamental set of solutions $\psi_{1}(x;C_{3},C_{4})$
and $\psi_{2}(x;C_{3},C_{4})$ of (56) as follows:

$u(x)=-2\ln\left(C_{2}\psi_{1}(x;C_{3},C_{4})+\psi_{2}(x;C_{3},C_{4})\right)+C_{1},$
(57)

where $C_{i}\in\mathbb{R}$ for $i=1,2,3,4.$

####  5.2.1 Some particular solutions in terms of elementary functions

For some particular values of the arbitrary constants in (57), the general
solution to equation (52) can be expressed in terms of elementary functions.
This is the case, for instance, when $C_{3}=C_{4}=0.$ For these particular
values, the Schrödinger-type equation (56) turns out to be simply
$\psi^{\prime\prime}=0$ and therefore a corresponding fundamental set of
solutions is given by $\psi_{1}(x)=1$ and $\psi_{2}(x)=x.$

In this case, the expression (57) provides the following two-parameter familiy
of exact solutions for equation (44):

$u(x)=-2\ln(x+C_{2})+C_{1},\quad C_{1},C_{2}\in\mathbb{R}.$

###  5.3 A fourth-order ODE without Lie point symmetries

This example illustrates the success of the $\mathcal{C}^{\infty}$-structure-
based method in solving ODEs for which the classical Lie method cannot be
applied due to the absence of Lie point symmetries in the equation. This is
the case of the fourth-order ODE

$u_{4}=-uu_{1}+uu_{3}+3u_{1}u_{2}+x+u_{2},$ (58)

whose associated vector field is

$Z=\partial_{x}+u_{1}\partial_{u}+u_{2}\partial_{u_{1}}+u_{3}\partial_{u_{2}}+\left(-uu_{1}+uu_{3}+3u_{1}u_{2}+x+u_{2}\right)\partial_{u_{3}}.$

It can be checked that the determining equations for a Lie point symmetry of
equation (58), in the form $v=\xi(x,u)\partial_{x}+\eta(x,u)\partial_{u}$,
yield the trivial solution $\xi=\eta=0$. Therefore equation (58) does not
admit Lie point symmetries.

Finding a $\mathcal{C}^{\infty}$structure for $\mathcal{S}(\\{Z\\})$ can be
simplified by assuming that some of the infinitesimals of the corresponding
elements are constant or linear with respect to $u_{1}$ and $u_{2}$. This is
similar to the approach used in the previous examples. In this way we find the
ordered set $\langle X_{1},X_{2},X_{3},X_{4}\rangle$ given by the following
vector fields:

$\begin{array}[]{l}X_{1}=\partial_{u}+u\partial_{u_{1}}+(u^{2}+u_{1})\partial_{u_{2}}+(u^{3}+3uu_{1}+u_{2})\partial_{u_{3}},\\\
X_{2}=\partial_{u_{1}}+u\partial_{u_{2}}+(u^{2}+2u_{1})\partial_{u_{3}},\\\
X_{3}=\partial_{u_{2}}+(u-1)\partial_{u_{3}},\\\
X_{4}=\partial_{u_{3}}.\end{array}$

It can be checked that the vector field $X_{1}$ is a
$\mathcal{C}^{\infty}$-symmetry of $\mathcal{S}(\\{Z\\})$, since
$[X_{1},Z]=uX_{1}$. On the other hand, the vector field $X_{2}$ is a symmetry
of $\mathcal{S}(\\{Z,X_{1}\\})$, because

$\begin{array}[]{l}[X_{2},Z]=X_{1},\\\ [X_{2},X_{1}]=0.\\\ \end{array}$

Finally, $X_{3}$ is a $\mathcal{C}^{\infty}$-symmetry of
$\mathcal{S}(\\{Z,X_{1},X_{2}\\})$, since the following commutation relations
are satisfied:

$\begin{array}[]{l}[X_{3},Z]=X_{2}-X_{3},\\\ [X_{3},X_{1}]=0,\\\
[X_{3},X_{2}]=0.\\\ \end{array}$

Thus, in accordance with Definition 3.2, and considering the pointwise linear
independence of $X_{1},X_{2},X_{3},X_{4}$, the ordered set $\langle
X_{1},X_{2},X_{3},X_{4}\rangle$ forms a $\mathcal{C}^{\infty}$-structure for
$\mathcal{S}({Z})$.

In what follows, we employ the integration method outlined in Subsection 3.1
to achieve our objective of solving equation (58). The corresponding 1-forms
provided in (5) yield the following expression when using
$\boldsymbol{\Omega}=dx\wedge du\wedge du_{1}\wedge du_{2}\wedge du_{3}:$

$\begin{array}[]{cl}\omega_{1}=&u_{1}dx-du,\\\ \omega_{2}=&(uu_{1}-u_{2})dx-
udu+du_{1},\\\
\omega_{3}=&-(uu_{2}+u_{1}^{2}-u_{3})dx+u_{1}du+udu_{1}-du_{2},\\\
\omega_{4}=&-(-uu_{1}-uu_{2}-u_{1}^{2}+x+u_{2}+u_{3})dx-(u_{1}+u_{2})du-(u+2u_{1})du_{1}\\\
&+(1-u)du_{2}+du_{3}.\end{array}$ (59)

The results obtained after applying the integration procedure, as described in
Subsection 3.1, are presented below.

  1. 1.

The Pfaffian equation $\omega_{4}\equiv 0$ is completely integrable and a
function $I_{4}=I_{4}(x,u,u_{1},u_{2},u_{3})$ such that
$dI_{4}\wedge\omega_{4}=0$ can be chosen as

$I_{4}=(-u_{1}^{2}-u_{1}u-(u-1)u_{2}+x+u_{3}+1)e^{-x}.$

  2. 2.

The restriction of the 1-form $\omega_{3}$ given in (59) to $\Sigma_{(C_{4})}$
provides

$\omega_{3}|_{\Sigma_{(C_{4})}}=(C_{4}e^{x}+u_{1}u-x-
u_{2}-1)dx+u_{1}du+udu_{1}-du_{2}.$

A function $I_{3}=I_{3}(x,u,u_{1},u_{2};C_{4})$ such that
$dI_{3}\wedge\omega_{3}|_{\Sigma_{(C_{4})}}=0$ is

$I_{3}=-\frac{1}{2}C_{4}e^{2x}-e^{x}(uu_{1}-x-u_{2}).$

  3. 3.

We now restrict the 1-form $\omega_{2}$ in (59) to $\Sigma_{(C_{3},C_{4})},$
resulting in

$\omega_{2}|_{\Sigma_{(C_{3},C_{4})}}=-\left(\frac{1}{2}C_{4}e^{x}-x+C_{3}e^{-x}\right)dx-
udu+du_{1}.$

A function $I_{2}=I_{2}(x,u,u_{1};C_{3},C_{4})$ such that
$dI_{2}\wedge\omega_{3}|_{\Sigma_{(C_{3},C_{4})}}=0$ can be calculated by
simple quadrature:

$I_{2}=-\frac{1}{2}C_{4}e^{x}+\frac{1}{2}x^{2}-\frac{1}{2}u^{2}+u_{1}+C_{3}e^{-x}.$

  4. 4.

Finally, the restriction of the 1-form $\omega_{1}$ given in (59) to
$\Sigma_{(C_{2},C_{3},C_{4})}$ turns out to be:

$\omega_{1}|_{\Sigma_{(C_{2},C_{3},C_{4})}}=\left(\frac{1}{2}u^{2}+\frac{1}{2}C_{4}e^{x}-\frac{1}{2}x^{2}-C_{3}e^{-x}+C_{2}\right)dx-
du.$ (60)

The integration of the Pfaffian equation
$\omega_{1}|_{\Sigma_{(C_{2},C_{3},C_{4})}}\equiv 0$ is equivalent to solve
the following first-order ODE:

$u_{1}=\frac{1}{2}u^{2}+\frac{1}{2}C_{4}e^{x}-\frac{1}{2}x^{2}-C_{3}e^{-x}+C_{2},$
(61)

which is of Riccati-type. Equation (61) can be mapped into the following
Schrödinger-type equation by means of the standard transformation
$u=-2\psi^{\prime}(x)/\psi(x)$:

$\psi^{\prime\prime}(x)=\left(\frac{1}{4}x^{2}-\frac{1}{4}C_{4}e^{x}+\frac{1}{2}C_{3}e^{-x}-\frac{1}{2}C_{2}\right)\psi(x).$
(62)

Therefore, if $\psi_{1}=\psi_{1}(x;C_{2},C_{3},C_{4})$ and
$\psi_{2}=\psi_{2}(x;C_{2},C_{3},C_{4})$ form a fundamental set of solutions
to equation (62), a first integral $I_{1}=I_{1}(x,u;C_{2},C_{3},C_{4})$ for
the Riccati equation (61) is given by [55, Proposition 4.1]

$I_{1}=\dfrac{u\psi_{1}(x;C_{2},C_{3},C_{4})+2\psi_{1}^{\prime}(x;C_{2},C_{3},C_{4})}{u\psi_{2}(x;C_{2},C_{3},C_{4})+2\psi_{2}^{\prime}(x;C_{2},C_{3},C_{4})}$
(63)

By setting $I_{1}(x,u;C_{2},C_{3},C_{4})=C_{1},$ where $C_{1}\in\mathbb{R},$
we obtain the general solution for equation (58), expressed in terms of a
fundamental set of solutions to equation (62):

$u(x)=\frac{-2\left(C_{1}\psi_{1}^{\prime}(x;C_{2},C_{3},C_{4})+\psi_{2}^{\prime}(x;C_{2},C_{3},C_{4})\right)}{C_{1}\psi_{1}(x;C_{2},C_{3},C_{4})+\psi_{2}(x;C_{2},C_{3},C_{4})},$
(64)

where $C_{i}\in\mathbb{R}$, for $i=1,2,3,4.$ In consequence, the
$\mathcal{C}^{\infty}$-structure approach successfully solves the fourth-order
equation (58), despite the absence of Lie point symmetries.

####  5.3.1 Some families of exact solutions in terms of special functions

For particular values of the constants $C_{2},C_{3}$ and $C_{4}$ appearing in
(62), a fundamental set of solutions to the Schrödinger-type equation (62) can
be expressed in terms of well-known special functions. For instance, when
$C_{3}=C_{4}=0,$ equation (62) becomes

$\psi^{\prime\prime}(x)=\left(\frac{1}{4}x^{2}-\frac{1}{2}C_{2}\right)\psi(x),$

which admits the following linearly independent solutions:

$\psi_{1}(x;C_{2})=\dfrac{\mathbf{W}_{\frac{1}{4}C_{2},\frac{1}{4}}\left(\frac{x^{2}}{2}\right)}{\sqrt{x}},\qquad\psi_{2}(x;C_{2})=\dfrac{\mathbf{M}_{\frac{1}{4}C_{2},\frac{1}{4}}\left(\frac{x^{2}}{2}\right)}{\sqrt{x}},$
(65)

where $\mathbf{M}_{\mu,\nu}=\mathbf{M}_{\mu,\nu}(z)$ and
$\mathbf{W}_{\mu,\nu}=\mathbf{W}_{\mu,\nu}(z)$ denotes the corresponding
Whittaker functions [56], i.e., two linearly independent solutions to the
equation

$\phi^{\prime\prime}(z)+\left(\dfrac{-1}{4}+\dfrac{\mu}{z}+\dfrac{\dfrac{1}{4}-\nu^{2}}{z^{2}}\right)\phi(z)=0.$

Therefore, a two-parameter family of solutions that corresponds to (64) when
$C_{3}=C_{4}=0,$ is given by

$u(x)=\frac{-x^{2}+C_{2}+1}{x}+\frac{4C_{1}\mathbf{W}_{\frac{1}{4}C_{2}+1,\frac{1}{4}}\left(\frac{1}{2}x^{2}\right)-\left(C_{2}+3\right)\mathbf{M}_{\frac{1}{4}C_{2}+1,\frac{1}{4}}\left(\frac{1}{2}x^{2}\right)}{x\left(C_{1}\mathbf{W}_{\frac{1}{4}C_{2},\frac{1}{4}}\left(\frac{1}{2}x^{2}\right)+\mathbf{M}_{\frac{1}{4}C_{2},\frac{1}{4}}\left(\frac{1}{2}x^{2}\right)\right)}.$
(66)

Since Whittaker functions can be defined in terms of hypergeometric or Kummer
functions, the family of solutions (66) could have alternatively been
expressed using other special functions. Furthermore, by selecting different
values for $C_{2}$ in (66), we can generate 1-parameter families of solutions
that involve various types of special functions, such as the following
examples:

  * •

For $C_{2}=0,$ (66) provides the next 1-parameter family of exact solutions

$u(x)=-x\frac{C_{1}\mathbf{I}_{-\frac{3}{4}}\left(\frac{1}{4}x^{2}\right)-\mathbf{K}_{\frac{3}{4}}\left(\frac{1}{4}x^{2}\right)}{C_{1}\mathbf{I}_{\frac{1}{4}}\left(\frac{1}{4}x^{2}\right)+\mathbf{K}_{\frac{1}{4}}\left(\frac{1}{4}x^{2}\right)},$
(67)

where $\mathbf{I}_{\nu}$ and $\mathbf{K}_{\nu}$ denote the modified Bessel
functions of the first and second kinds, respectively.

  * •

When $C_{2}=-1,$ we obtain the following 1-parameter family of exact solutions
for equation (52):

$u(x)=-x+\frac{4e^{-\frac{1}{2}x^{2}}}{2C_{1}-\sqrt{2\pi}\mbox{Erf}\left(\frac{1}{2}\sqrt{2}x\right)},$
(68)

where Erf denotes the error function (54).

##  6 Concluding remark

In this work it has been demonstrated the effectiveness of the
$\mathcal{C}^{\infty}$-structure procedure as a novel tool to deal with
integrability problems in differential equations. By applying the integration
method based on $\mathcal{C}^{\infty}$-structures, several models have been
fully integrated, including a Lotka-Volterra model and equations for which the
Lie method encounters certain obstacles when trying to obtain the general
solution.

Consequently, $\mathcal{C}^{\infty}$-structures offer significant
contributions to solving problems that cannot be solved by classical methods,
expanding our understanding and analytical capabilities in tackling intricate
mathematical problems.

##  7 Acknowledgments

The authors thank the finantial support from Junta de Andalucía to the
research group FQM–377 and from Universidad de Cádiz through "Plan Propio de
Estímulo y Apoyo a la Investigación y Transferencia 2022/2023". A. Ruiz and C.
Muriel thank the partial financial support by the grant "Operator Theory: an
interdisciplinary approach", reference ProyExcel$\\_$00780, a project financed
in the 2021 call for Grants for Excellence Projects, under a competitive
bidding regime, aimed at entities qualified as Agents of the Andalusian
Knowledge System, in the scope of the Andalusian Research, Development and
Innovation Plan (PAIDI 2020). Counseling of University, Research and
Innovation of the Junta de Andalucía.

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