1 Introduction

While General Relativity is a well-tested theory of gravity, it cannot provide a physical mechanism to the problems which follow from cosmological observations [1, 2]. Nowadays, cosmologists deal with alternative and modified theories of gravity by keeping the geometric character of the gravitational theory [3, 4]. With the term teleparallelism [5] we describe a family of gravitational theories where the fundamental geometric object is the teleparallel connection [6] related to the nonholonomic frame, in contrast to General Relativity, where the fundamental the geometric object is the Levi–Civita connection related to the metric tensor.

As in the case of General Relativity, in which the gravitational Lagrangian is defined by the Ricci scalar R of the Levi–Civita connection, in teleparallel geometry, the torsion scalar T is considered for the definition of Action Integral. When the gravitational Action Integral is linear in the torsion scalar T, the theory is equivalent to general relativity, and it is known as the Teleparallel Equivalent of General Relativity (TEGR) [7,8,9]. However, the equivalence ends here because when one introduces the gravitational action, scalar fields that are coupled to gravity [10,11,12,13] or nonlinear terms of the geometric scalars [14,15,16,17], the two approaches give different results. The latter is easy to understand because the Ricci scalar includes second-order derivatives of the metric tensor, while the torsion scalar T includes only first-order derivatives for the arbitrary functions of the nonholonomic frame.

On the other hand, scalar fields play a significant role in describing the matter component responsible for the different phases of the universe during the cosmic evolution [18, 19]. Indeed, the early accelerated era of the universe known as inflation is attributed to the inflaton scalar field [20,21,22]. Furthermore, scalar fields have been used as unified dark energy models that can describe the late-time acceleration phase of the universe and the dark matter component [23,24,25]. Scalar fields provide a simple mathematical mechanism for introducing additional degrees of freedom in the field equations, consequently enriching the cosmological dynamics and evolution. Thus, multiscalar field models in gravity have been widely considered by cosmologists over the last years; see, for instance, [26,27,28,29,30,31] and references therein.

In this study, we are interested in a multiscalar field cosmological model in the context of scalar-torsion theory. Scalar-torsion theory is the analogue of the scalar–tensor theory in teleparallelism in which the scalar field is non-minimally coupled to gravity with interaction in the Action Integral between the scalar field and the torsion scalar T [10,11,12, 32,33,34,35,36,37,38,39,40]. Another important characteristic of the scalar-torsion theory is that the scalar field can attribute the additional degrees of freedom to a higher-order teleparallel theory [13], similar to Horndeski gravity [41] or to the relation of the O’Hanlon theory with \(f\left( R\right) \)-gravity [4]. Furthermore, we consider the existence of a second scalar field minimally coupled to gravity but with an interacting term in the kinetic part with the scalar-torsion field. Specifically, we define a model similar to that studied before in scalar–tensor theory in [42]. This family of models can provide the hyperbolic inflation epoch [27] in the Jordan frame [43]. For the background space, we assume the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric, which admits six isometries. We consider the scalar fields to inherit the symmetries of the background space from where it follows that the gravitational field equations are reduced into ordinary differential equations. Moreover, the study of the evolution and the asymptotic dynamics of the physical parameters has been widely applied in many physical theories to construct criteria that a proposed theory should satisfy to be cosmologically viable [44,45,46,47,48,49,50]. Furthermore, the Noether symmetry analysis determines conservation laws for the field equations and infers the integrability properties [51, 52]. Conservation laws are applied for the construction of analytic solutions. Some applications in the literature can be found in [53,54,55,56,57,58,59] and references therein.

The structure of the paper is as follows: Sect. 2 presents the cosmological model of our consideration, which is a two-scalar field model in the context of teleparallelism. Specifically, we consider the scalar-torsion theory with a second scalar field minimally coupled to gravity but with an interaction between the two scalar fields. We assume that the interaction between the two scalar fields is provided by the potential function and by the coupling function of the scalar-torsion model. The field equations form a Hamiltonian system with a minisuperspace description and a point-like Lagrangian exist. Hence, Noether’s theorem is applied in Sect. 3 to constrain the unknown parameters of this multiscalar field model. We find seven families of potential functions where non-trivial conservation laws exist. In Sect. 4, we use the Noether symmetry classification to determine exact solutions and Liouville integrable cosmological models to derive analytic solutions in a multiscalar field cosmological model in the teleparallel theory of gravity. The qualitative behaviour of the solutions is discussed. In Appendix A, we extend the Noether symmetry analysis for nonzero spatial curvature for the background geometry. Finally, in Sect. 5, we summarize our results and conclude.

2 Multiscalar-torsion cosmology

We consider the gravitational action integral in teleparallelism

$$\begin{aligned} S= & {} \frac{1}{16\pi G}\int \textrm{d}^{4}xe\Bigg [ F\left( \phi \right) \left( T+\frac{\omega }{2}\phi _{;\mu } \phi ^{\mu }\right. \nonumber \\{} & {} \quad \left. +V\left( \phi \right) +L\left( x^{\kappa },\psi ,\psi _{;\mu } \right) \right) \Bigg ]. \end{aligned}$$
(1)

where T is the torsion scalar defined by the antisymmetric Weitzenböck connection \(\hat{\Gamma }^{\lambda }{}_{\mu \nu }\), that is,

$$\begin{aligned} T={S_{\beta }}^{\mu \nu }{T^{\beta }}_{\mu \nu }, \end{aligned}$$

in which \(T_{\mu \nu }^{\beta }\) is the torsion tensor defined as \(T_{\mu \nu }^{\beta }=\hat{\Gamma }_{\nu \mu }^{\beta }-\hat{\Gamma }_{\mu \nu }^{\beta }\) and \({S_{\beta }}^{\mu \nu }=\frac{1}{2}({K^{\mu \nu }}_{\beta }+\delta _{\beta }^{\mu }{T^{\theta \nu }}_{\theta }-\delta _{\beta }^{\nu }{T^{\theta \mu }}_{\theta })\) where now \({K^{\mu \nu }}_{\beta }=-\frac{1}{2}({T^{\mu \nu }}_{\beta } -{T^{\nu \mu }}_{\beta }-{T_{\beta }}^{\mu \nu })~\)is the contorsion tensor. We remark that the Weitzenböck connection \(\hat{\Gamma }^{\lambda }{}_{\mu \nu }\) is related to the vierbein fields \(e_{i}=h_{i}^{\mu } \partial _{i}\), as \(\hat{\Gamma }^{\lambda }{}_{\mu \nu }=h_{a}^{\lambda } \partial _{\mu }h_{\nu }^{a}\) with the metric tensor \(g_{\mu \nu } ~\)to be defined as \(g_{\mu \nu }=\eta _{ij}h_{\mu }^{i}h_{\nu }^{j}\). As mentioned above, we work with the Weitzenböck connection instead of the teleparallel one, where the spin connection \(\omega ^a{}_{b\mu }\) vanishes.

The Lagrangian density \(L\left( x^{\kappa },\psi ,\psi _{;\mu } \right) \) is considered to describe the dynamics of a second-scalar field, that is,

$$\begin{aligned} L\left( x^{\kappa },\psi ,\psi _{;\mu } \right) =\frac{\beta }{2}\psi _{;\mu }\left( x^{\kappa }\right) \psi ^{;\mu }\left( x^{\kappa }\right) . \end{aligned}$$
(2)

Hence, the gravitational Action Integral (1) reads

$$\begin{aligned} S&=\frac{1}{16\pi G}\int \textrm{d}^{4}xe\Bigg [ F\left( \phi \right) \Bigg ( T+\frac{\omega }{2}\phi _{;\mu } \phi ^{\mu }\nonumber \\&\quad +V\left( \phi \right) +\frac{\beta }{2}\psi _{;\mu } \psi ^{;\mu } \Bigg ) +\hat{V}\left( \phi ,\psi \right) \Bigg ]\,, \end{aligned}$$
(3)

where the potential function \(\hat{V}\left( \phi ,\psi \right) \) has been introduced to describe the interaction between the two scalar fields.

At large scales, the universe is assumed to be described by the spatially flat FLRW geometry with line element

$$\begin{aligned} \textrm{d}s^{2}=-\textrm{d}t^{2}+a^{2}(t)(\textrm{d}x^{2}+\textrm{d}y^{2}+\textrm{d}z^{2}). \end{aligned}$$
(4)

where \(a\left( t\right) \) is the scale factor.

A proper set of vierbein fields where the limit of General Relativity is recovered for \(\phi =\) constant and \(\psi =\) constant, is the following

$$\begin{aligned} h_{~\mu }^{i}(t)=\textrm{diag}(1,a(t),a(t),a(t)), \end{aligned}$$
(5)

from which we calculate the torsion scalar \(T=6H^{2},\) with \(H=\frac{\dot{a}}{a}\), \(\dot{a}=\frac{\textrm{d}a}{\textrm{d}t}\), to be the Hubble function.

By replacing the latter expression for the torsion scalar in the action Integral (3) and assume that the scalar fields inherit the isometries of the FLRW universe, we derive the point-like Lagrangian

$$\begin{aligned} \mathcal {L}\left( a,\dot{a},\phi ,\dot{\phi },\psi ,\dot{\psi }\right)= & {} F\left( \phi \right) \left( 6a\dot{a}^{2}+a^{3}\left( \frac{\omega }{2}\dot{\phi }^{2}+\frac{\beta }{2}\dot{\psi }^{2}\right) \right. \nonumber \\{} & {} \left. +a^{3}V\left( \phi ,\psi \right) \right) . \end{aligned}$$
(6)

where \(\hat{V}\left( \phi ,\psi \right) =V\left( \phi ,\psi \right) F\left( \phi \right) . ~\)

The gravitational field equations read

$$\begin{aligned}{} & {} 6H^{2}+\frac{\omega }{2}\dot{\phi }^{2}+\frac{\beta }{2}\dot{\psi }^{2}-V\left( \phi ,\psi \right) =0, \end{aligned}$$
(7)
$$\begin{aligned}{} & {} \dot{H}+\frac{3}{2}H^{2}-\left( \frac{1}{4}\left( \frac{\omega }{2}\dot{\phi }^{2}+\frac{\beta }{2}\dot{\psi }^{2}+V\left( \phi \right) \right) \right. \nonumber \\{} & {} \qquad \qquad \left. -H\dot{\phi }\left( \ln \left( F\left( \phi \right) \right) \right) _{,\phi }\right) =0, \end{aligned}$$
(8)
$$\begin{aligned}{} & {} \ddot{\phi }+3H\dot{\phi }+\frac{1}{\omega }\left( \ln \left( F\left( \phi \right) \right) \right) _{,\phi }\left( \frac{\omega }{2}\dot{\phi } ^{2}-\frac{\beta }{2}\dot{\psi }^{2}-6H^{2}\right. \nonumber \\{} & {} \qquad \qquad \left. -V\left( \phi ,\psi \right) \right) -\frac{1}{\omega }V_{,\phi }=0, \end{aligned}$$
(9)

and

$$\begin{aligned} \ddot{\psi }+3H\dot{\psi }+\left( \ln \left( F\left( \phi \right) \right) \right) _{,\phi }\dot{\phi }\dot{\psi }-\frac{1}{\beta }V_{,\psi }=\,0. \end{aligned}$$
(10)

The dynamical system of second-order ordinary differential equations is autonomous and admits as conservation law the constraint equation (7) which can be seen as the Hamiltonian function

$$\begin{aligned} \mathcal {H}\equiv F\left( \phi \right) \left( 6a\dot{a}^{2}{+}a^{3}\left( \frac{\omega }{2}\dot{\phi }^{2}{+}\frac{\beta }{2}\dot{\psi }^{2}\right) {-}a^{3}V\left( \phi ,\psi \right) \right) =0. \nonumber \\ \end{aligned}$$
(11)

The evolution of the cosmological variables depends on the nature of the scalar field potential \(V\left( \phi ,\psi \right) \). The scope of this work is to define the functional forms of \(V\left( \phi ,\psi \right) \) by using the Noether symmetry analysis. Specifically, we shall perform a classification of the potential function \(V\left( \phi ,\psi \right) \) such that the point-like Lagrangian to admit non-trivial Noether symmetries, that is, non-trivial conservation laws. Such analysis has been the subject of study in various cosmological models.

The novelty of this approach is two-fold. The determination of conservation laws for the field equations are essential for the study of the integrability properties of a given theory or for the construction of invariant functions which can describe the dynamical evolution of the physical explicitly variables in a specific region in the space of solutions. On the other hand, Noether symmetries of the field equations are related to the collineations of the minisuperspace; that is, it is the geometry of the minisuperspace which impose the existence of Noether symmetries and the specific forms of the potential function \(V\left( \phi ,\psi \right) \). Hence, the Noether symmetry analysis is a geometric approach to analyzing gravitational models.

3 Noether symmetry analysis

Let us briefly discuss the symmetries of differential equations and present the Noether’s two theorems for one-parameter point transformations.

Consider a system of second-order differential equations

$$\begin{aligned} \ddot{y}^{A}=\omega ^{A}\left( t,y^{B},\dot{y}^{B}\right) , \end{aligned}$$
(12)

where t is the independent variable and \(y^{A}\) are the dependent variables; in our cosmological model \(y^{A}=\left( a,\phi ,\psi \right) \).

Let the function \(\Phi \) be the map of one parameter point transformations \(\Phi \) \(:\left\{ t,y^{B}\right\} \rightarrow \left\{ \bar{t}\left( t,y^{B},\varepsilon \right) ,\bar{y}^{B^{\prime }}\left( t,y^{B},\varepsilon \right) \right\} \), where \(\varepsilon \) is an infinitesimal parameter. Function \(\Phi \,\) maps solutions of the system (12) into solutions if and only if

$$\begin{aligned} X^{\left[ 2\right] }\left( \ddot{y}^{A}-\omega ^{A}\left( t,y^{B},\dot{y}^{B}\right) \right) =0, \end{aligned}$$
(13)

where \(X^{\left[ 2\right] }\) is the second extension of the vector field

$$\begin{aligned} X=\xi \left( t,y^{B},\varepsilon \right) \Bigg |_{\varepsilon =0}\partial _{t}+\eta ^{A}\left( t,y^{B},\varepsilon \right) \Bigg |_{\varepsilon =0}\partial _{A} \end{aligned}$$
(14)

with

$$\begin{aligned} \xi \left( t,y^{B},\varepsilon \right) =\frac{\partial \bar{t}}{\partial \varepsilon },~\eta ^{A}\left( t,y^{B},\varepsilon \right) =\frac{\partial \bar{y}^{A}}{\partial \varepsilon }. \end{aligned}$$
(15)

Therefore, the second extension is given by \(X^{\left[ 2\right] } =X+\eta ^{\left[ 1\right] A}\partial _{A}+\eta ^{\left[ 2\right] A} \partial _{A}\), in which \(\eta ^{\left[ 1\right] A}=\dot{\eta }^{A}-\dot{y} ^{A}\dot{\xi }\) and \(\eta ^{\left[ 2\right] A}=\dot{\eta }^{\left[ 1\right] A}-\ddot{y}^{A}\dot{\xi }\). Finally, when the symmetry condition (13) is true, the generator X will be called a Lie point symmetry for the dynamical system (12).

For dynamical systems which follow from a variational principle, Emmy Noether published in a pioneer work two theorems which relate the symmetries of the differential equations to the variational symmetries and the existence of invariant functions, which are conservation laws.

Consider now the Action Integral \(S=\int \mathcal {L}\left( t,y^{A},\dot{y} ^{A}\right) \textrm{d}t\) which describes the dynamical system (12), where \(\mathcal {L}\left( t,y^{A},\right. \left. \dot{y}^{A}\right) \) is the Lagrange function. Noether’s first theorem states that a Lie symmetry for the dynamical system (12) is also a variational symmetry for the Action Integral S, if and only if there exist a function \(f\left( t,y^{A},\ldots \right) \) such that the following condition to be true

$$\begin{aligned} X^{\left[ 1\right] }\mathcal {L}+\mathcal {L}\dot{\xi }=\dot{f}. \end{aligned}$$
(16)

If the latter condition is true, the vector field X is characterized as Noether symmetry. The function f is a boundary term introduced to allow for the infinitesimal changes in the value of the Action Integral produced by an infinitesimal change in the boundary of the domain caused by the transformation of the variables in the Action Integral.

Noether’s second theorem provides a simple and systematic way for the construction of conservation laws for each Noether symmetry. Indeed, if X is a given Noether symmetry for the dynamical system (12) described by the Lagrange function \(\mathcal {L}\left( t,y^{A},\dot{y}^{A}\right) \), then, quantity

$$\begin{aligned} I=\left( \dot{y}^{A}\frac{\partial \mathcal {L}}{\partial \dot{y}^{A} }-\mathcal {L}\right) \xi -\frac{\partial \mathcal {L}}{\partial \dot{y}^{A}} \eta ^{A}+f. \end{aligned}$$
(17)

is a conservation law, that is \(\dot{I}=0\).

For dynamical systems described by Lagrangian functions of the form

$$\begin{aligned} \mathcal {L}\left( t,y^{A},\dot{y}^{A}\right) =\frac{1}{2}\gamma _{AB}\left( y^{C}\right) \dot{y}^{A}\dot{y}^{B}-U\left( y^{C}\right) , \end{aligned}$$
(18)

it has been found that Noether symmetries are constructed by the elements of the Homothetic algebra of the second-rank tensor, i.e. the metric tensor, \(\gamma _{AB}\left( y^{C}\right) \).

Lagrangian function (6) consists of two unknown functions, function \(F\left( \phi \right) \), which defines the geometry of the minisuperspace and the effective potential \(V_\textrm{eff}\left( a,\phi ,\psi \right) =a^{3}F\left( \phi \right) V\left( \phi ,\psi \left( x^{\kappa }\right) \right) \). Hence, by following the analysis described in [52], the Noether symmetry analysis is two-fold: firstly, we shall classify the functional forms of \(F\left( \phi \right) \) where the minisuperspace of (6) admits Homothetic vector fields, secondly, the homothetic vectors will be used to constraint the functional form of the effective potential \(V_\textrm{eff}\left( \phi ,\psi \right) \) and write the corresponding Noether symmetry and conservation law.

3.1 Noether symmetry classification

Consider now that \(F\left( \phi \right) \) is a non-constant function. Then, the classification of the Homothetic algebra for the minisuperspace gives three cases, \(F_{A}\left( \phi \right) \) is arbitrary, \(F_{B}\left( \phi \right) =F_{0}e^{2K\phi },\) where K is an arbitrary non-zero constant; and \(F_{C}\left( \phi \right) =F_{0}e^{K\phi }\), with \(K=\pm \frac{\sqrt{3\left( -\omega \right) }}{2}\).

For \(F_{A}\left( \phi \right) \) the Homothetic algebra of the minisuperspace has two dimensions, and it consists of the Killing vector

$$\begin{aligned} K^{1}=\partial _{\psi }, \end{aligned}$$
(19)

and the proper Homothetic vector

$$\begin{aligned} Y=\frac{2}{3}a\partial _{a}. \end{aligned}$$
(20)

On the other hand, for functions \(F_{B}\left( \phi \right) \) and \(F_{C}\left( \phi \right) \) the Homothetic algebras admitted by the minisuperspace are of four and five dimensions, respectively. Indeed, for \(F_{B}\left( \phi \right) \) the additional Killing vector fields are

$$\begin{aligned} K^{2}=-\frac{2}{3}Ka\psi \partial _{a}+\psi \partial _{\phi }+\frac{8K\ln a-\omega \phi }{\beta }\partial _{\psi }, \end{aligned}$$
(21)

and

$$\begin{aligned} K^{3}=-\frac{2}{3}Ka\partial _{a}+\partial _{\phi }. \end{aligned}$$
(22)

Furthermore, when \(K=+\frac{\sqrt{3\left( -\omega \right) }}{4}\), that is, in the case \(F_{C}\left( \phi \right) \), the fourth Killing vector field of the minisuperspace is

$$\begin{aligned} K^{4}= & {} \left( -\frac{2}{3}a-\frac{a\sqrt{-3\omega }}{6}\phi +a\ln a\right) \partial _{a}\nonumber \\{} & {} +\left( \phi -\frac{6}{\sqrt{-3\omega }}\ln a\right) \partial _{\phi }+\psi \partial _{\psi }\text {.} \end{aligned}$$
(23)

At this point we remark that the case \(K=-\frac{\sqrt{3\left( -\omega \right) }}{2}\) is recovered under the change of variables \(\phi \rightarrow -\phi \).

As far as the classification of the potential function \(V\left( \phi ,\psi \right) \) is concerned for each case of the coupling function \(F\left( \phi \right) \) follows.

For arbitrary function \(F_{A}\left( \phi \right) \), and arbitrary potential there exist the trivial Noether symmetry \(\partial _{t}\) with corresponding conservation law the constraint condition (7).

For \(V_{1}\left( \phi ,\psi \right) =V\left( \phi \right) \), the field equations admit the Noether symmetry vector \(K^{1}\) with corresponding conservation law

$$\begin{aligned} I^{1}\left( K^{1}\right) =a^{3}e^{2K\phi }\beta \dot{\psi }. \end{aligned}$$
(24)

Moreover, for \(V_{2}\left( \phi ,\psi \right) =e^{-4\delta \psi }V\left( \phi \right) \), there exists the Noether symmetry \(\delta \left( 2t\partial _{t}+Y\right) +K^{1}\), with conservation law

$$\begin{aligned} I^{2}\left( \delta \left( 2t\partial _{t}+Y\right) +K^{1}\right) = 2\delta t\mathcal {H}-e^{2K\phi }a^{2}\left( 8\delta \dot{a}+\beta a\dot{\psi }\right) .\nonumber \\ \end{aligned}$$
(25)

For the exponential function \(F_{B}\left( \phi \right) \) there exist additional functional forms of the potential where Noether symmetries exist. Indeed, when \(V_{3}\left( \phi ,\psi \right) =V\left( \psi \right) \) the vector field \(K^{3}\) is a Noether symmetry with conservation law

$$\begin{aligned} I^{3}\left( K^{3}\right) {=}-e^{2K\phi }a^{2}\left( 8K\dot{a}-\omega a\dot{\phi }\right) \,. \end{aligned}$$
(26)

When \(V_{4}\left( \phi ,\psi \right) =V\left( \psi -\frac{\phi }{\alpha }\right) \), there exist the Noether symmetry \(K^{1}+\alpha K^{3}\) with conservation law

$$\begin{aligned} I^{4}\left( \alpha K^{1}+K^{3}\right) =I^{1}\left( K^{1}\right) +\frac{1}{\alpha }I^{3}\left( K^{3}\right) . \end{aligned}$$
(27)

Furthermore, for \(V_{5}\left( \phi ,\psi \right) =V\left( \alpha \psi -\phi \right) e^{-\bar{\delta }\psi },~\bar{\delta }=\frac{\delta }{a}\), there exist the additional Noether symmetry is the vector field \(\delta \left( 2t\partial _{t}+Y\right) +K^{1}+\alpha K^{3}\) where now the Noether conservation law is

$$\begin{aligned} I^{5}\left( \delta \left( 2t\partial _{t}+Y\right) +K^{1}+\alpha K^{3}\right)= & {} I^{2}\left( \delta \left( 2t\partial _{t}+Y\right) +K^{1}\right) \nonumber \\{} & {} +\alpha I^{3}\left( K^{3}\right) . \end{aligned}$$
(28)

For \(V_{6}\left( \phi ,\psi \right) =V_{0}\), there exist the additional Noether symmetry \(K^{2}\) with corresponding conservation law

$$\begin{aligned} I^{6}\left( K^{2}\right)= & {} e^{2K\phi }a^{2}\left( -8Ka\psi \dot{a}+\omega a\psi \dot{\phi }\right. \nonumber \\{} & {} \left. +a\left( 8K\ln a-\omega \phi \right) \dot{\psi }\right) . \end{aligned}$$
(29)

When \(V_{7}\left( \phi ,\psi \right) =V\left( \psi \right) e^{-4\delta \phi },~\) the vector field \(\delta \left( 2t\partial _{t}\right. \left. +Y\right) +K^{3}\) is a Noether symmetry with conservation law

$$\begin{aligned} I^{7}\left( \delta \left( 2t\partial _{t}+Y\right) +K^{3}\right) =2\delta t\mathcal {H}-8\delta e^{2K\phi }a^{2}\dot{a}-I^{3}\left( K^{3}\right) . \end{aligned}$$

Finally, for the third function form \(F_{C}\left( \phi \right) \), despite the existence of more elements in the Homothetic algebra of the minisuperspace there are not any other functional forms of the potential function \(V\left( \phi ,\psi \right) \) where additional Noether symmetries exist.

A natural question that occurs from the above analysis is, whether we can conclude about the integrability properties of one of the above potential functions or if we can define invariant functions such as to determine exact solutions. The dependent variables define a three-dimensional space in which the evolution of the physical variables takes place. Hence, a specific cosmological model will be called Liouville integrable if there exist at least three conservation laws which are independent and are in involution. Easily from the previous results, it follows that this is true for the constant potential \(V_{6}\left( \phi ,\psi \right) =V_{0}\) and for the exponential potential \(V_{7}\left( \phi ,\psi \right) =V_{0}e^{-4\delta \phi }\), where at least the conservations laws \(\left\{ \mathcal {H},I^{1}\left( K^{1}\right) ,I^{3}\left( K^{3}\right) \right\} ,\) and \(\left\{ \mathcal {H},I^{1}\left( K^{1}\right) ,I^{7}\left( \delta \left( 2t\partial _{t}+Y\right) +K^{3}\right) \right\} \) are independent and in involution, assuming the constraint condition (11), \(\mathcal {H}\equiv 0\).

4 Exact and analytic solutions

In this section, we present some exact closed-form solutions for the field equations, as well as the analytic solution for the Liouville integrable cosmological model. Indeed, we consider the exponential coupling function \(F\left( \phi \right) =F_{0}e^{2K\phi }\) where without loss of generality we select parameter \(K=1\).

4.1 Exact solutions

Consider now the exponential potential \(V\left( \phi ,\psi \right) =V_{0}e^{-\delta _{1}\psi }e^{-\delta _{2}\phi }\). From the previous analysis it is clear that, for this specific potential function, the cosmological point-like Lagrangian (6) admits as Noether symmetries the vector fields \(\left\{ \delta \left( 2t\partial _{t}+Y\right) +K^{1},K^{1}+\frac{\delta _{2}}{\delta _{1}}K^{3}\right\} \) and any linear combination of these two. From these vectors, it is easy to construct the invariant functions

$$\begin{aligned} a\left( t\right) =a_{0}t^{a_{1}},~~\phi \left( t\right) =\phi _{1}\ln t~\text {and}~\psi \left( t\right) =\psi _{1}\ln t,~ \end{aligned}$$
(30)

where \(2-4\delta _{2}\phi _{1}-4\delta _{1}\psi _{1}=0\). By replacing these in the field equations, we end with an algebraic system that gives the following solutions

$$\begin{aligned} a_{1}&=\frac{1-\phi _{1}}{3},~ \end{aligned}$$
(31)
$$\begin{aligned} V_{0}&=0, \end{aligned}$$
(32)
$$\begin{aligned} \omega&=-\frac{16\delta _{1}^{2}\left( \phi _{1}-1\right) ^{3} +3\beta \left( 1-2\delta _{2}\phi _{1}\right) }{12\delta _{1}^{2}\phi _{1}^{2}}, \end{aligned}$$
(33)

or

$$\begin{aligned} a_{1}&=\frac{2-\phi _{1}}{3},~\end{aligned}$$
(34)
$$\begin{aligned} V_{0}&=\frac{\beta \left( 2\delta _{2}\phi _{1}-1\right) }{8\delta _{1}^{2}},~\end{aligned}$$
(35)
$$\begin{aligned} \omega&=\frac{8\delta _{1}^{2}\left( 2-\phi _{1}\right) +3\beta \delta _{2}\left( 1-2\delta _{2}\phi _{1}\right) }{6\delta _{1}^{2}\phi _{1}}, \end{aligned}$$
(36)

or

$$\begin{aligned} a_{1}&=\frac{\beta \left( 2\delta _{2}\beta _{1}-1\right) }{16\delta _{1}^{2}},~\end{aligned}$$
(37)
$$\begin{aligned} V_{0}&=\frac{\beta \left( 2\delta _{2}\phi _{1}-1\right) \left( 16\delta _{1}^{2}\left( \phi _{1}-1\right) +\beta \left( 6\delta _{2}\phi _{1}-3\right) \right) }{128\delta _{1}^{4}},~\end{aligned}$$
(38)
$$\begin{aligned} \omega&=-\frac{\beta \left( 2\delta _{2}-1\right) \left( 2\delta _{2} \phi _{1}-1\right) }{4\delta _{1}^{2}\phi _{1}}. \end{aligned}$$
(39)

We remark that solutions with \(\phi _{1}\psi _{1}=0\) are not accepted because in this case, at least one of the scalar fields do not contribute in the cosmological fluid.

On the other hand, when \(\delta _{1}=0\), that is \(\phi _{1}=\frac{1}{2\delta _{2}}\), we end with the exact solution

$$\begin{aligned} a_{1}=\frac{1}{3}\left( 1-\frac{1}{2\delta _{2}}\right) ,~V_{0}=0\text { and }\nonumber \\ \omega =-\frac{4}{3}\left( 1+\delta _{2}\left( \delta _{2}\left( 4+3\beta \psi _{1}^{2}\right) -4\right) \right) . \end{aligned}$$
(40)

That is a scaling solution, which means that an ideal gas describes the cosmological fluid with the equation of state parameter \(w_\textrm{eff}\), where \(w_\textrm{eff}=\frac{2}{3a_{1}}-1\), or \(w_\textrm{eff}=\frac{2\delta _{1}+1}{2\delta _{1}-1} \). Thus, acceleration occurs when \(-\frac{1}{4}<\delta _{1}<\frac{1}{2}\).

4.2 Analytic solution for vanishing potential

For the zero potential function \(V\left( \phi ,\psi \right) =0,~\)we define the new dependent variable \(u=3\ln a+\phi \) where the point-like Lagrangian (6) reads

$$\begin{aligned} \mathcal {L}\left( u,\dot{u},\phi ,\dot{\phi },\psi ,\dot{\psi }\right) =e^{u+\phi }\left( \left( 4+3\omega \right) \dot{\phi }^{2}-8\dot{\phi }\dot{u}\right. \nonumber \\ \left. +4\dot{u}^{2}+3\beta \dot{\psi }^{2}\right) . \end{aligned}$$
(41)

We define the canonical momentum components

$$\begin{aligned} p_{u}=\frac{\partial \mathcal {L}}{\partial \dot{u}}, \quad p_{\phi }=\frac{\partial \mathcal {L}}{\partial \dot{\phi }}, \quad p_{\psi }=\frac{\partial \mathcal {L}}{\partial \dot{\psi }}, \end{aligned}$$
(42)

that is

$$\begin{aligned} \dot{u}= & {} \frac{1}{4\omega }e^{-u-\phi }\left( \left( 4+3\omega \right) p_{u}+4p_{\phi }\right) , \end{aligned}$$
(43)
$$\begin{aligned} \dot{\phi }= & {} \frac{1}{\omega }e^{-u-\phi }\left( p_{u}+p_{\phi }\right) , \end{aligned}$$
(44)
$$\begin{aligned} \dot{\psi }= & {} \frac{1}{\beta }e^{-u-\phi }p_{\psi }. \end{aligned}$$
(45)

Hence, we can write the Hamiltonian as follows

$$\begin{aligned} \mathcal {H}\equiv \frac{1}{8\omega \beta }e^{-u-\phi }\left( 4\left( p_{u}{+}p_{\phi }\right) ^{2}\beta {+}4\omega p_{\psi }^{2}{+}3p_{u}^{2}\omega \beta \right) =0. \nonumber \\ \end{aligned}$$
(46)

The field equations are (43)–(45) and

$$\begin{aligned} \dot{p}_{u}=0, \quad \dot{p}_{\phi }=0,\quad \dot{p}_{\psi }=0. \end{aligned}$$
(47)

Hence \(p_{u},~p_{\phi }\) and \(p_{\psi }\) are integration constants. Consequently, the field equations are of the form

$$\begin{aligned} \dot{u}=u_{1}e^{-u-\phi },\quad \dot{\phi }=\phi _{1}e^{-u-\phi }, \quad \dot{\psi }=\psi _{1}e^{-u-\phi }. \end{aligned}$$

In order to write the analytic solution in closed-form expression, we perform the change of variables \(\frac{\textrm{d}u}{\textrm{d}t}=\frac{\textrm{d}u}{\textrm{d}\tau }\frac{\textrm{d}\tau }{\textrm{d}t}\), with \(\textrm{d}\tau =e^{-u-\phi }\textrm{d}t\) or \(\textrm{d}t=e^{u+\phi }\textrm{d}\tau \). Hence, the field equations are

$$\begin{aligned} \frac{\textrm{d}u}{\textrm{d}\tau }=u_{1}, \quad \frac{\textrm{d}\phi }{\textrm{d}\tau }=\phi _{1} \quad \text {and} \quad \frac{\textrm{d}\psi }{\textrm{d}\tau }=\psi _{1}, \end{aligned}$$
(48)

that is

$$\begin{aligned} u\left( t\right) =u_{1}\tau +u_{0},\quad \phi =\phi _{1}\tau +\phi _{0} \quad \text {and} \quad \psi =\psi _{1}\tau +\psi _{0}\text {.} \end{aligned}$$

Hence, the line element for the physical space is of the form

$$\begin{aligned} \textrm{d}s^{2}=-e^{2\left( u_{1}+\phi _{1}\right) \tau }\textrm{d}\tau +e^{\frac{2}{3}\left( u_{1}-\phi _{1}\right) t}\left( \textrm{d}x^{2}+\textrm{d}y^{2}+\textrm{d}z^{2}\right) ,\nonumber \\ \end{aligned}$$
(49)

Moreover, we derive the relation \(t=\frac{1}{\left( u_{1}+\phi _{1}\right) }e^{\left( u_{1}+\phi _{1}\right) \tau }\),or \(\tau =\frac{1}{\left( u_{1}+\phi _{1}\right) }\ln \left( \left( u_{1}+\phi _{1}\right) t\right) \). Thus, the line element for the FLRW spacetime in the lapse function \(N\left( t\right) =1\), becomes

$$\begin{aligned} \textrm{d}s^{2}=-\textrm{d}t+\left( \left( u_{1}+\phi _{1}\right) t\right) ^{\frac{2}{3} \frac{\left( u_{1}-\phi _{1}\right) }{\left( u_{1}+\phi _{1}\right) } }\left( \textrm{d}x^{2}+\textrm{d}y^{2}+\textrm{d}z^{2}\right) .\nonumber \\ \end{aligned}$$
(50)

The latter solution describes an ideal gas solution with the constant equation of state parameter \(w_\textrm{eff}=-1+2\frac{u_{1}+\phi _{1}}{u_{1}-\phi _{1}}~\), from where we infer that the solution describes an accelerated universe when \(\frac{u_{1}+\phi _{1}}{u_{1}-\phi _{1}}<\frac{1}{3}\).

4.3 Analytic solution for constant potential

Consider now the constant potential \(V\left( \phi ,\psi \right) =V_{0}\). The solution process is similar to before. We define the new variable \(U=3\ln a+2\phi \), and in terms of Hamiltonian formalism, the field equations are

$$\begin{aligned} \dot{U}= & {} \frac{1}{4\omega }e^{-U}\left( \left( 16+3\omega \right) p_{U}+8p_{\phi }\right) ,\\ \dot{\phi }= & {} \frac{1}{\omega }e^{-U}\left( 2p_{U}+p_{\phi }\right) ,\nonumber \\ \dot{\psi }= & {} \frac{1}{\beta }e^{-U}p_{\psi },\nonumber \end{aligned}$$
(51)

and

$$\begin{aligned}{} & {} \dot{p}_{U}=\frac{1}{4\beta \omega }e^{-U}\left( 4\left( 2p_{U}+p_{\phi }\right) ^{2}\beta +4\omega p_{\psi }^{2}+3\beta \omega p_{U}^{2}\right) ,\nonumber \\ \end{aligned}$$
(52)
$$\begin{aligned}{} & {} \dot{p}_{\phi }=0,~\dot{p}_{\psi }=0, \end{aligned}$$
(53)

where the constraint equation is

$$\begin{aligned} \left( 16{+}3\omega \right) p_{U}^{2}{+}\frac{4\omega }{\beta }p_{\psi }^{2} {+}16p_{U}p_{\phi }{+}4\left( p_{\phi }^{2}{-}2e^{2U}\omega V_{0}\right) =0\text {.}\nonumber \\ \end{aligned}$$
(54)

Consequently, the conservation laws for the field equations are the momentum \(p_{\phi }\) and \(p_{\psi }\).

We proceed with the derivation of the Action \(S\left( U,\phi ,\psi \right) \) by solving the Hamilton-Jacobi equation

$$\begin{aligned}{} & {} \left( 16+3\omega \right) \left( \frac{\partial S}{\partial U}\right) ^{2}+\frac{4\omega }{\beta }\left( \frac{\partial S}{\partial \psi }\right) ^{2}+16\left( \frac{\partial S}{\partial U}\right) \left( \frac{\partial S}{\partial \phi }\right) \nonumber \\{} & {} \quad +4\left( \left( \frac{\partial S}{\partial \phi }\right) ^{2}-2e^{2U}\omega V_{0}\right) =0\text {,} \end{aligned}$$
(55)

where the conservation laws give \(\left( \frac{\partial S}{\partial \phi }\right) -p_{\phi 0}=0\) and \(\left( \frac{\partial S}{\partial \psi }\right) -p_{\psi 0}=0\). Thus, it follows

$$\begin{aligned} S\left( U,\phi ,\psi \right) =S_{1}\left( U\right) +p_{\phi 0}\phi +p_{\psi 0}\psi , \end{aligned}$$

where \(S_{1}\left( U\right) \) is given by the first-order ordinary differential equation

$$\begin{aligned}{} & {} \frac{dS_{1}\left( U\right) }{dU}\nonumber \\{} & {} =\frac{2\sqrt{\omega \left( \left( 16{+}3\omega \right) \left( 2e^{2A}V_{0}\beta {-}p_{\psi 0}^{2}\right) {-}3\beta p_{\phi 0}^{2}\right) }{-}8\sqrt{\beta }p_{\phi 0}}{\sqrt{\beta }\left( 16{+}3\omega \right) }\text {.}\nonumber \\ \end{aligned}$$
(56)

Therefore, the field equations are reduced in the following system

$$\begin{aligned} \dot{U}= & {} \frac{1}{4\omega }e^{-U}\left( \left( 16+3\omega \right) \frac{dS_{1}\left( U\right) }{dU}+8p_{\phi 0}\right) , \end{aligned}$$
(57)
$$\begin{aligned} \dot{\phi }= & {} \frac{1}{\omega }e^{-U}\left( 2\frac{dS_{1}\left( U\right) }{dU}+p_{\phi 0}\right) , \end{aligned}$$
(58)
$$\begin{aligned} \dot{\psi }= & {} \frac{1}{\beta }e^{-U}p_{\psi 0}. \end{aligned}$$
(59)

The analytic solution is expressed as follows

$$\begin{aligned} e^{U\left( t\right) }= & {} \frac{\left( \exp \left( \sqrt{\frac{V_{0}\left( 16+3\omega \right) }{2\omega }}t\right) +2V_{0}\beta \left( 16+3\omega \right) \left( 3p_{\phi 0}^{2}\beta +p_{\psi 0}^{2}\left( 16+3\omega \right) \right) \exp \left( -\sqrt{\frac{V_{0}\left( 16+3\omega \right) }{2\omega }}t\right) \right) }{4V_{0}\beta \left( 16+3\omega \right) },\quad \qquad \end{aligned}$$
(60)
$$\begin{aligned} \phi \left( t\right)= & {} \phi _{1}+\frac{2\sqrt{\omega }}{\sqrt{V_{0} 16+3\omega }}U\left( t\right) \nonumber \\{} & {} +\frac{4\left( 3p_{\phi 0}\sqrt{\omega \beta }\right) }{\left( 16+3\omega \right) \sqrt{3\beta p_{\phi 0}^{2}+p_{\psi 0}^{2}\left( 16+3\omega \right) }}\arctan \left( \frac{\exp \left( \sqrt{\frac{V_{0}\left( 16+3\omega \right) }{2\omega }}t\right) }{\sqrt{2V_{0}\beta \left( 16+3\omega \right) \left( 3\beta p_{\phi 0}^{2}+p_{\psi 0}^{2}\left( 16+3\omega \right) \right) }}\right) ,\qquad \qquad \end{aligned}$$
(61)

and

$$\begin{aligned} \psi \left( t\right) =\frac{p_{\psi 0}}{\beta }\int e^{-U}dt\text {~}. \end{aligned}$$
(62)

We conclude that in order for the solution to be real, the following constraints follow \(\frac{V_{0}\left( 16+3\omega \right) }{2\omega }>0\), and \(3\beta p_{\phi 0}^{2}+p_{\psi 0}^{2}\left( 16+3\omega \right) >0\). In the late universe, the asymptotic behaviour of the analytic solution is

$$\begin{aligned} e^{U\left( t\right) }\simeq \exp \left( \sqrt{\frac{V_{0}\left( 16+3\omega \right) }{2\omega }}t\right) , \quad \phi \left( t\right) \simeq U\left( t\right) . \end{aligned}$$

that is, \(a\simeq e^{a_{1}t},\) from where it follows that the de Sitter Universe describes the late-time evolution of this specific cosmological model.

4.4 Analytic solution for exponential potential

In the case of the exponential potential \(V\left( \phi \right) =V_{0} e^{-4\delta \phi }\) we proceed with the definition of the new dependent variable \(A=3\ln a+2\left( 1-\delta \right) \phi \). The field equations are written in the Hamiltonian formalism

$$\begin{aligned} e^{A+2\phi }\dot{A}=\frac{1}{4\omega }\left( 3\omega p_{A}+8\left( p_{\phi }+2p_{A}\left( 1-\delta \right) \right) \left( 1-\delta \right) \right) ,\nonumber \\ \end{aligned}$$
(63)
$$\begin{aligned}{} & {} e^{A+2\phi }\dot{\phi }=\frac{1}{\omega }\left( p_{\phi }+2\left( 1-\delta \right) p_{A}\right) ,~\nonumber \\{} & {} e^{A+2\phi }\dot{\psi }=\frac{1}{\beta }p_{\psi }, \end{aligned}$$
(64)
$$\begin{aligned} e^{A+2\phi }\dot{p}_{A}= & {} \frac{1}{4\omega \beta }\left( 4\beta \left( p_{\phi }+2p_{A}\left( 1-\delta \right) \right) ^{2}\right. \nonumber \\{} & {} \left. +4\omega p_{\psi }^{2} +3\beta \omega p_{A}^{2}\right) ,\nonumber \\ \dot{p}_{\phi }= & {} 0, \quad \dot{p}_{\psi }=0. \end{aligned}$$
(65)

with constraint equation

$$\begin{aligned}{} & {} 4p_{\psi }^{2}\omega +\beta \left( 16p_{A}p_{\phi }\left( 1-\delta \right) +p_{A}^{2}\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) \right. \nonumber \\{} & {} \quad \left. +4\left( p_{\phi }^{2}-2V_{0}\omega e^{2A}\right) \right) =0. \end{aligned}$$
(66)

Consequently, \(p_{\phi }\left( t\right) =p_{\phi 0}\) and \(p_{\psi }\left( t\right) =p_{\psi 0}\) are the two conservation laws.

We write the Hamilton–Jacobi equation and, with the use of the conservation laws, we derive the following functional form for the Action

$$\begin{aligned} S\left( A,\phi ,\psi \right) =S_{1}\left( A\right) +p_{\phi 0}\phi +p_{\psi 0}\psi , \end{aligned}$$
(67)

where now

$$\begin{aligned} \frac{dS_{1}\left( A\right) }{dS}=\frac{8p_{\phi 0}\sqrt{\beta }\left( 1-\delta \right) +2\sqrt{\omega \left( \left( 2e^{2A}\beta V_{0}-p_{\psi 0}^{2}\right) \left( 16\left( 1-\delta \right) ^{2}+3\omega \right) -3\beta p_{\phi 0}^{2}\right) }}{\sqrt{\beta }\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) }, \end{aligned}$$
(68)
Table 1 Classification according to the Noether Symmetry analysis. In the first two cases, the coupling function is an arbitrary \(F(\phi )\), and in the rest, it is \(F_B(\phi ) = F_0 e^{2K\phi }\) with K being a constant. The proper Homothetic vector Y is given by Eq. (20)
Full size table

Hence, by replacing with \(p_{A}=\frac{dS_{1}\left( A\right) }{dS}\) we end with a system of three first-order differential equations. We can now derive the scalar fields \(\phi ,~\psi \) as function of A, that is,

$$\begin{aligned} \phi \left( A\right)= & {} \phi _{1}{-}\frac{8\left( 1{-}\delta \right) }{16\left( 1{-}\delta \right) ^{2}{+}3\omega }A {-}\frac{6p_{\phi 0}\sqrt{\omega \beta }\arctan \left( \frac{\sqrt{2e^{A}\beta V_{0}\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) {-}\left( 3p_{\phi 0}^{2}\beta {+}p_{\psi 0}^{2}\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) \right) }}{\sqrt{3p_{\phi 0}^{2}\beta {+}p_{\psi 0} ^{2}\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) }}\right) }{\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) \left( \sqrt{3p_{\phi 0}^{2}\beta +p_{\psi 0}^{2}\left( 16\left( 1-\delta \right) ^{2}{+}3\omega \right) }\right) },\qquad \qquad \end{aligned}$$
(69)
$$\begin{aligned} \psi \left( A\right)= & {} \frac{2p_{\psi 0}^{2}\sqrt{\omega }\arctan \left( \frac{\sqrt{2e^{A}\beta V_{0}\left( 16\left( 1{-}\delta \right) ^{2} {+}3\omega \right) {-}\left( 3p_{\phi 0}^{2}\beta {+}p_{\psi 0}^{2}\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) \right) }}{\sqrt{3p_{\phi 0}^{2} \beta {+}p_{\psi 0}^{2}\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) } }\right) }{\sqrt{\beta \left( 3p_{\phi 0}^{2}\beta {+}p_{\psi 0}^{2}\left( 16\left( 1{-}\delta \right) ^{2}{+}3\omega \right) \right) }}\text {. } \end{aligned}$$
(70)

In order to write the analytic solution in closed-form expression in terms of the independent variable, we make the change of variable \(dt=e^{A+2\phi }d\tau \). Thus, in terms of \(\tau \) the analytic solution reads

$$\begin{aligned}{} & {} e^{A\left( \tau \right) }=\sqrt{\frac{3p_{\phi 0}^{2}\beta +p_{\psi 0} ^{2}\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) }{2\beta V_{0}\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) }}\left( \cos \left( \frac{1}{2}\sqrt{\frac{3p_{\phi 0}^{2}\beta +p_{\psi 0}^{2}\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) }{\beta \omega }}\tau \right) \right) ^{-2}\\{} & {} \phi \left( \tau \right) =\phi _{1}+\frac{3p_{\phi 0}\sqrt{\omega \beta } -8\sqrt{\beta }\left( 1-\delta \right) \sqrt{\omega }\ln \left( \cos \left( \frac{1}{2}\sqrt{\frac{3p_{\phi 0}^{2}\beta +p_{\psi 0}^{2}\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) }{\beta \omega }}\tau \right) \right) }{2\sqrt{\omega \beta }\left( 16\left( 1-\delta \right) ^{2}+3\omega \right) },\nonumber \end{aligned}$$
(71)

and

$$\begin{aligned} \psi \left( \tau \right) =-\frac{p_{\psi 0}}{\beta }\tau \text {.} \end{aligned}$$

It is easy to see that the de Sitter universe can be recovered as a late-time attractor.

5 Conclusions

In this work, we considered a two scalar field cosmology, where one of the fields couples with a dilatonic coupling to gravity, while the second one couples minimally; there is, however, an interacting potential between the two fields. As explained earlier, there is a strong motivation for using multiple scalar fields in cosmology since it can provide much richer phenomenology with both inflationary and late-time dark energy models. Many studies have considered more than one scalar field in curvature gravity, but in the teleparallel geometry, where torsion is responsible for the gravitational forces instead of curvature, the works performed are limited.

We performed a symmetry analysis to classify invariant models under point transformations. We find seven classes of the coupling function and the interaction potential for which non-trivial conservation laws exist. The results are summarized in Table 1. In the first column, we have the form of the coupling function \(F(\phi )\); in the second one, the form of the potential \(V(\phi ,\psi )\); in the third one, the symmetry vector and in the last one the corresponding conservation law.

The symmetry analysis would mean nothing if it could not help us find exact solutions for the system. For example, we have shown that when the coupling function is of the form \(F(\phi ) = F_0 e^{2 \phi }\), we can find some closed-form analytic solutions for the scale factor and the scalar fields for four different configurations of the interacting potential \(V(\phi ,\psi )\). Therefore, in follow-up work, we plan to study the stability of these solutions.