  In the attempts to apply Finsler geometry to construct an extension of
general relativity, the question about a suitable generalization of the
Einstein equations is still under debate. Since Finsler geometry is based on a
scalar function on the tangent bundle, the field equation which determines this
function should also be a scalar equation. In the literature two such equations
have been suggested: the one by Rutz and the one by one of the authors. Here we
employ the method of canonical variational completion to show that Rutz
equation can not be obtained from a variation of an action and that its
variational completion yields the latter field equations. Moreover, to improve
the mathematical rigor in the derivation of the Finsler gravity field equation,
we formulate the Finsler gravity action on the positive projective tangent
bundle. This has the advantage of allowing us to apply the classical
variational principle, by choosing the domains of integration to be compact and
independent of the dynamical variable. In particular in the pseudo-Riemannian
case, the vacuum field equation becomes equivalent to the vanishing of the
Ricci tensor.
