We address the issue of mobility of localized modes in two-dimensional
nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes
e.g. discrete spatial solitons in a tight-binding approximation of
two-dimensional optical waveguide arrays made from photorefractive crystals. We
discuss numerically obtained exact stationary solutions and their stability,
focussing on three different solution families with peaks at one, two, and four
neighboring sites, respectively. When varying the power, there is a repeated
exchange of stability between these three solutions, with symmetry-broken
families of connecting intermediate stationary solutions appearing at the
bifurcation points. When the nonlinearity parameter is not too large, we
observe good mobility, and a well defined Peierls-Nabarro barrier measuring the
minimum energy necessary for rendering a stable stationary solution mobile.Comment: 19 pages, 4 figure