The sine-lattice models derive from an hamiltonian where both the coupling potential and the on-site potential are periodic functions with multiple degenerate minima. This allows the existence of a new class of localized topological excitations, the rotating modes, which share however some properties on the non topological oscillatory modes. An approximate analytical solution is obtained. Numerical simulations show that an exact solution is likely to exist and that the rotating modes are stable against collisions with breathers and can be thermally generated. Due to their structure, these modes which exist for any value of the coupling, strong or weak, are intrinsically discrete. Some consequences for physics are discussed.