Meromorphic extension of solutions of soliton equations

We consider local versions of the direct and inverse scattering transforms and describe their analytic properties, which are analogous to the properties of the classical Laplace and Borel transforms. This enables us to study local holomorphic solutions of those integrable equations on $\mathbb C^2_{xt}$ whose complexified forms are given by the zero curvature condition for connections of the form $U\,dx+V\,dt$, where $U$ is a linear function of the spectral parameter $z$ and $V$ is a polynomial of degree $m\geqslant2$ in $z$. We show that the local holomorphic Cauchy problem for such equations is soluble if and only if the scattering data of the initial condition belong to Gevrey class $1/m$. We also show that every local holomorphic solution extends to a global meromorphic function of $x$ for every fixed $t$.