  In this work we explain the relevance of the Differential Galois Theory in
the semiclassical (or WKB) quantification of some two degree of freedom
potentials. The key point is that the semiclassical path integral
quantification around a particular solution depends on the variational equation
around that solution: a very well-known object in dynamical systems and
variational calculus. Then, as the variational equation is a linear ordinary
differential system, it is possible to apply the Differential Galois Theory to
study its solvability in closed form. We obtain closed form solutions for the
semiclassical quantum fluctuations around constant velocity solutions for some
systems like the classical Hermite/Verhulst, Bessel, Legendre, and Lam\'e
potentials. We remark that some of the systems studied are not integrable, in
the Liouville-Arnold sense.
