In this paper we compute all the smooth solutions to theHamilton-Jacobi equation associated with the horocycle flow. The latter can be seen as the Euler-Lagrange flow at energy  of  the Tonelli Lagrangian  given by (hyperbolic) kinetic energy plus standard magnetic potential. The method we use is to look at Lagrangian graphs that are contained in the level set  where  denotes the Hamiltonian dual to L.

dynamical systems, Hamilton-Jacobi equation, magnetic flows.