# Integrability from Point Symmetries in a family of Cosmological Horndeski Lagrangians

## TL;DR

This paper investigates the integrability of Horndeski cosmological models using point symmetries and Noether's theorem, deriving solutions like De Sitter and ideal gas universes, and exploring conformal transformations.

## Contribution

It identifies point symmetries in Horndeski models, derives conservation laws, and connects different cosmological solutions through symmetry analysis and conformal transformations.

## Key findings

- Derivation of De Sitter and ideal gas solutions using symmetry invariants
- Identification of conservation laws via Noether's theorem in Horndeski models
- Analysis of conformal transformations linking different cosmological solutions

## Abstract

For a family of Horndeski theories, formulated in terms of a generalized Galileon model, we study the integrability of the field equations in a Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime. We are interested in point transformations which leave invariant the field equations. Noether's theorem is applied to determine the conservation laws for a family of models that belong to the same general class. The cosmological scenarios with or without an extra perfect fluid with constant equation of state parameter are the two important cases of our study. \ The De Sitter universe and ideal gas solutions are derived by using the invariant functions of the symmetry generators as a demonstration of our result. Furthermore, we discuss the connection of the different models under conformal transformations while we show that when the Horndeski theory reduces to a canonical field the same holds for the conformal equivalent theory. Finally we discuss how singular solutions provides nonsingular universes in a different frame and vice versa.