  We study stationary configurations mimicking nonholonomic locally anisotropic
black rings (for instance, with ellipsoidal polarizations and/or imbedded into
solitonic backgrounds) in three/six dimensional pseudo-Finsler/ Riemannian
spacetimes. In the asymptotically flat limit, for holonomic configurations, a
subclass of such spacetimes contains the set of five dimensional black ring
solutions with regular rotating event horizon. For corresponding
parameterizations, the metrics and connections define Finsler-Einstein
geometries modeled on tangent bundles, or on nonholonomic (pseudo) Riemannian
manifolds. In general, there are vacuum nonholonomic gravitational
configurations which can not be generated in the limit of zero cosmological
constant.
