Integrable evolutionary equations and their holomorphic solutions

An evolutionary equation (that is, a vector field on the set of functions of the spatial variable along with their jets) is said to be integrable if there is an infinite set of linearly independent vector fields which formally commute with it. Examples are given by linear (heat), linearizable (Burgers) and soliton (Korteweg-de Vries) equations. This talk is devoted to studying analytical properties of local holomorphic solutions of such equations. We discuss analytic continuation to a globally meromorphic function of the spatial variable, the Painlevé property, trivial monodromy of solutions of the auxiliary linear problem, the place of finite-gap solutions among all local holomorphic solutions, and pole dynamics.