Summary: Stationary solutions of higher order KdV equations play an important role for the study of the KdV equation itself. They give rise to the coefficients of the associated Lax pair $(P,L)$ for which $P$ and $L$ have an algebraic relationship (and are therefore called algebro-geometric). This paper gives a sufficient condition for rational and simply periodic functions which are bounded at infinity to be algebro-geometric as those potentials of $L$ for which $Ly=zy$ has only meromorphic solutions. It also gives a new elementary proof that this is a necessary condition for any meromorphic function to be algebro-geometric.

35Q53, 34A20, 58F07

KdV equation, algebro-geometric solutions of integrable systems, meromorphic solutions of linear differential equations