Picard potential and Hill's equation on a torus (Q1372998)

The paper deals with an intimate connection between Picard's theorem and elliptic finite-gap solutions of completely integrable systems. The authors show that elliptic finite-gap potentials of Hill's equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variables complementing a classical theorem of Picard. Moreover, the authors construct the hyperelliptic Riemann surface associated with a finite-gap potential (not necessarily elliptic) which is elegant and new.