A system of nearest neighbors Kuramoto-like coupled oscillators placed in a ring is studied above the critical synchronization transition. We find a richness of solutions when the coupling increases, which exists only within a solvability region (SR). We also find that the solutions possess different characteristics, depending on the section of the boundary of the SR where they appear. We study the birth of these solutions and how they evolve when the coupling strength increases, and determine the diagram of solutions in phase space.