We show that actin lamellar fragments driven solely by polymerization forces at the bounding membrane are generically motile when the circular symmetry is spontaneously broken, with no need of molecular motors or global polarization. We base our study on a nonlinear analysis of a recently introduced minimal model [Callan-Jones et al., Phys. Rev. Lett. 100, 258106 (2008)]. We prove the nonlinear instability of the center of mass and find an exact and simple relation between shape and center-of-mass velocity. A complex subcritical bifurcation scenario into traveling solutions is unfolded, where finite velocities appear through a nonadiabatic mechanism. Examples of traveling solutions and their stability are studied numerically.