We study the ground state of a generalized Frenkel-Kontorova model with a transverse degree of freedom. The model describes a lattice of atoms with a fixed concentration, interacting by long-range repulsive forces, which is submitted to a two-dimensional substrate potential periodic (sinusoidal) in one direction and symmetric (parabolic) or asymmetric (Toda-like) in the transverse direction. When the magnitude of the interatomic repulsion increases, the ground state of the model undergoes a series of bifurcations. In particular, the first bifurcation leads to a zigzag ground state and results in drastic change of system properties, including a cusp in the average elastic constant. For incommensurate cases, the bifurcation can interplay with the Aubry transition from a pinned to a sliding state. A reentrant pinned state has, for instance, been found. The nature (continuous or discontinuous) of the next bifurcations depends on the symmetry of the substrate potential in the transverse direction. Finally, we discuss briefly the applicability of the model to describe conductivity of superionic conductors, surface diffusion, and crystal growth.