  We consider an autonomous, indefinite Lagrangian admitting an infinitesimal
symmetry whose associated Noether charge is linear in each tangent space. Our
focus lies in investigating solutions to the Euler-Lagrange equations having
fixed energy and that connect a given point to a flow line of the infinitesimal
generator $K$. By utilizing the invariance of the Lagrangian under the flow of
$K$, we simplify the problem into a two-point boundary problem. Consequently,
we derive an equation that involves the differential of the ``arrival time'',
seen as a functional on the infinite dimensional manifold of connecting paths
satisfying the semi-holonomic constraint defined by the Noether charge. When
the Lagrangian is positively homogeneous of degree two in the velocities, the
resulting equation establishes a variational principle that extends the
Fermat's principle in a stationary spacetime. Furthermore, we also analyze the
scenario where the Noether charge is affine.
