We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) x R-2. We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary aD. The Bernoulli equation states that the "Bernoulli function" H :=-1/2 vertical bar v vertical bar(2) + p (where v is the velocity field and p the pressure) is constant along stream lines, that is, each particle is associated with a particular value of H. We also prescribe the value of H on partial derivative D. The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form v = del f x del g and deriving a degenerate nonlinear elliptic system for f and g. This system is solved using the Nash Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see, e.g., the book by Q. Han and J.-X. Hong (2006). Since we can allow H to be nonconstant on partial derivative D, our theory includes three-dimensional flows with nonvanishing vorticity.