It is discussed a locally anisotropic metric ansatz that, through a functional parameter, interpolates between local Ricci flat backgrounds (the Schwarzschild metric) and the expanding cosmological background (the flat Friedman-Robertson-Walker metric), hence generalizing the isotropic McVittie metric maintaining space-time free of extended singularities.

Possible physical interpretations for the metric functional parameter are discussed, in particular the covariant parameterization of Dark Matter effects within the Solar System and within galaxies.