A Frenkel-Kontorova model with a piecewise concave parabolic potential with downward cusps such that atoms could be pinned at the potential minima is exactly solved. An infinite series of first-order transitions, which may be understood as the dissociation of a large “molecule” into two smaller ones, is found as the strength of the potential increases from zero. The minimum energy configurations need not have a well-defined winding number. Given the winding number, the ground state configurations in general are highy degenerate and it is shown that one of them can be depicted by an increasing hull function. A novel type of non-recurrent minimum energy configuration, which may be viewed as defects carrying “fractional charge”, exists.