  We apply a perturbative technique to study classical Liouville theory on the
torus. After mapping the problem on the cut-plane we give the perturbative
treatment for a weak source. When the torus reduces to the square the problem
is exactly soluble by means of a quadratic transformation in terms of
hypergeometric functions. We give general formulas for the deformation of a
torus and apply them to the case of the deformation of the square. One can
compute the Heun parameter to first order and express the solution in terms of
quadratures. In addition we give in terms of quadratures of hypergeometric
functions the exact symmetric Green function on the square on the background
generated by a one point source of arbitrary strength.
