  In this paper we classify the solutions to the geometric Neumann problem for
the Liouville equation in the upper half-plane or an upper half-disk, with the
energy condition given by finite area. As a result, we classify the conformal
Riemannian metrics of constant curvature and finite area on a half-plane that
have a finite number of boundary singularities, not assumed a priori to be
conical, and constant geodesic curvature along each boundary arc.
