  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.
