  We perform an analytical investigation of the cell interface dynamics in the
framework of a minimal phase field model of cell motility suggested in [1],
which consists of two coupled evolution equations for the order parameter and a
two-dimensional vector field describing the actin network polarization
(orientation). We derive a closed evolutionary integro-differential equation
governing the cell interface dynamics. The equation includes the normal
velocity of the membrane, curvature, volume relaxation, and a parameter that is
determined by the non-equilibrium effects in the cytoskeleton. This equation
can be simplified to obtain a Burgers-like equation. A condition on the system
parameters for the existence of a stationary cell shape is obtained.
