Monodromy-free operators and holomorphic solutions of soliton equations

We consider the equations of isospectral deformation of Schrödinger or Dirac (Zakharov–Shabat) operators and show that all their holomorphic solutions are monodromy-free. This can be extended to higher-order oper- ators and systems and has consequences for the structure of holomorphic solutions of solution equations, which will also be discussed in the talk.