  We introduce a new model equation for Stokes gravity waves based on conformal
transformations of Euler's equations. The local version of the model equation
is relevant for dynamics of shallow water waves. It allows us to characterize
the traveling periodic waves both in the case of smooth and peaked waves and to
solve the existence problem exactly, albeit not in elementary functions.
Spectral stability of smooth waves with respect to co-periodic perturbations is
proven analytically based on the exact count of eigenvalues in a constrained
spectral problem.
