On the compatibility of supermetrics with nonlinear connections

One of the Helmholtz conditions for the inverse problem of a ‎Lagrangian Mechanics is the metric compatibility of a semispray‎ ‎and the associated nonlinear connection with a generalized‎ ‎Lagrange metric‎. ‎In this paper‎, ‎with respect to the supermetric‎ ‎induced by the Hessian of the Lagrangian‎, ‎we find a family of‎ ‎nonlinear connections compatible with supermetric‎. ‎In a particular case‎, ‎when a Lagrangian superfunction‎ ‎is regular‎, ‎we have a solution for the Euler-Lagrange‎ ‎superequation which‎ ‎defines a metric nonlinear connection‎.