  We study a two-dimensional free boundary problem that models motility of
eukaryotic cells on substrates. This problem consists of an elliptic equation
describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE
for the density of myosin motors. The two key properties of this problem are
(i) presence of the cross diffusion as in the classical Keller-Segel problem in
chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves
curvature of the boundary. We establish the bifurcation of the traveling waves
from a family of radially symmetric steady states. The traveling waves describe
persistent motion without external cues or stimuli which is a signature of cell
motility. We also prove existence of non-radial steady states. Existence of
both traveling waves and non-radial steady states is established via
Leray-Schauder degree theory applied to a Liouville-type equation (which is
obtained via a reduction of the original system) in a free boundary setting.
