Smooth Hamilton-Jacobi solutions for the Horocycle flow

In this paper we compute all the smooth solutions to the Hamilton-Jacobi equation associated with the horocycle flow. This can be seen as the Euler-Lagrange flow (restricted to the energy level set $E^{-1}(\frac 12)$) defined by the Tonelli Lagrangian $L:T\mathbb H\rightarrow \mathbb R$ given by (hyperbolic) kinetic energy plus the standard magnetic potential. The method we use is to look at Lagrangian graphs that are contained in the level set $\{H=\frac 12\}$, where $H:T^*\mathbb H\rightarrow \mathbb R$ denotes the Hamiltonian dual to $L$.