  In this paper we introduce a notion of integrability in the non autonomous
sense. For the cases of 1 + 1/2 degrees of freedom and quadratic homogeneous
Hamiltonians of 2 + 1/2 degrees of freedom we prove that this notion is
equivalent to the classical complete integrability of the system in the
extended phase space. For the case of quadratic homogeneous Hamiltonians of 2 +
1/2 degrees of freedom we also give a reciprocal of the Morales-Ramis result.
We classify those systems by terms of symplectic change of frames involving
algebraic functions of time, and give their canonical forms.
