  \We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
$u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted,
complementary classes of initial data.
