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For a family of Horndeski theories, formulated in terms of a generalizedGalileon model, we study the integrability of the field equations in aFriedmann-Lema\^{\i}tre-Robertson-Walker spacetime. We are interested in pointtransformations which leave invariant the field equations. Noether's theorem isapplied to determine the conservation laws for a family of models that belongto the same general class. The cosmological scenarios with or without an extraperfect fluid with constant equation of state parameter are the two importantcases of our study. \ The De Sitter universe and ideal gas solutions arederived by using the invariant functions of the symmetry generators as ademonstration of our result. Furthermore, we discuss the connection of thedifferent models under conformal transformations while we show that when theHorndeski theory reduces to a canonical field the same holds for the conformalequivalent theory. Finally we discuss how singular solutions providesnonsingular universes in a different frame and vice versa.

Integrability from Point Symmetries in a family of Cosmological Horndeski Lagrangians