Holomorphic solutions of soliton equations

We would like to present selected results concering the following 3 themes.
1) Around the Cauchy–Kovalevskaya theorem. (What can be said about analytic continuation and possible singularities of holomorphic solutions of general linear [or nonlinear] PDE's with constant {or, more generally, holomorphic] coefficients?)
2) The Riemann problem in the theory of integrable systems. (Construction of globally meromorphic solutions of soliton equations by means of the Riemann factorization problem for matrix-valued functions on a circle. Analytic properties of the resulting solutions.)
3) A local version of the inverse scattering method. (Criterion of solubility of the holomorphic Cauchy problem for soliton equations in terms of the scattering data of the initial condition. A description of the hull of meromorphy for any local holomorphic solution.)