  In this work we compute the families of classical Hamiltonians in two degrees
of freedom in which the Normal Variational Equation around an invariant plane
falls in Schroedinger type with polynomial or trigonometrical potential. We
analyze the integrability of Normal Variational Equation in Liouvillian sense
using the Kovacic's algorithm. We compute all Galois groups of Schroedinger
type equations with polynomial potential. We also introduce a method of
algebrization that transforms equations with transcendental coefficients in
equations with rational coefficients without changing essentially the Galoisian
structure of the equation. We obtain Galoisian obstructions to existence of a
rational first integral of the original Hamiltonian via Morales-Ramis theory.
