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\We consider an inverse scattering problem for Schr\"odinger operators withenergy dependent potentials. The inverse problem is formulated as aRiemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved fortwo distinct symmetry classes. As an application we prove global existencetheorems 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

A Riemann - Hilbert problem for an energy dependent Schrödinger operator