Holomorphic Cauchy problems for hierarchies of soliton equations

For every soliton equation there is an infinite sequence of higher-order soliton equations which commute with it as vector fields on the space of functions of the spatial variable $x$. A holomorphic germ $u_0(x)$ at $x_0$ is said to be admissible for an equation (labelled by the order $m$ of the highest $x$-derivative occurring in it) in this hierarchy if the Cauchy problem $u(x,t_0)=u_0(x)$ for this equation has a local holomorphic solution $u(x,t)$ in a neighbourhood of $(x_0,t_0)$. We prove that the set of germs admissible for the $m$-th flow consists of globally meromorphic functions of $x$ for every $m$ and strictly decreases as $m$ grows. We also establish the inclusions $\{$the germs of stationary (=finite-gap) solutions$\}\subset\{$the germs with convergent Baker–Akhiezer function$\}\subset\{$the germs admissible for all flows$\}$ and show that both inclusions are proper.