Singular Solution of the Liouville Equation under Perturbation

The Cauchy problem for the Liouville equation with a small perturbation is considered. We are interested in the asymptotics of the perturbed solution under the assumption that one has singularity. The main goal is to study both the asymptotic approximation of the singular lines and the asymptotic approximation of the solution everywhere outside the narrow neighbourhood of the singular lines.