A variational setting for an indefinite Lagrangian with an affine Noether charge

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold. Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function. Our setting includes different types of Lorentz-Finsler Lagrangians admitting a timelike Killing vector field.