  We report a novel algorithm of constructing linear and nonlinear potentials
in the two-dimensional Gross-Pitaevskii equation subject to given boundary
conditions, which allow for exact analytic solutions. The obtained solutions
represent superfluid flows in inhomogeneous Bose-Einstein condensates. The
method is based on the combination of the similarity reduction of the
two-dimensional Gross-Pitaevskii equation to the one-dimensional nonlinear
Schrodinger equation, the latter allowing for exact solutions, with the
conformal mapping of the given domain, where the flow is considered, to a
half-space. The stability of the obtained flows is addressed. A number of
stable and physically relevant examples are described.
