We are interested in finding equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Differential equations governing the mechanical equilibrium are derived using a continuum description of the membrane motions given by the surface Navier--Stokes equations with bending forces. Equilibrium conditions that are found appear to be independent of lateral viscosity and relate tension, pressure and tangential velocity of the fluid. These conditions yield that only surfaces with Killing vector fields, such as axisymmetric shapes, can support non-zero stationary flow of mass. We derive a shape equation that extends a classical Helfrich model with area constraint to membranes of non-negligible mass. We introduce a simple numerical method to compute solutions of this highly non-linear equation. The numerical method is then applied to find a diverse family of equilibrium configurations.