Lepton Flavor Violation: Constraints from exotic muon to electron conversion

The exotic neutrinoless $\mu^- - e^-$ nuclear conversion is studied within the conventional extensions of the standard model as well as in the minimal supersymmetric (SUSY) models with and without R-parity conservation. The dependence of the $\mu^- - e^-$ conversion rates on the nucleon and nuclear structure is consistently taken into account. Using our calculated transition matrix elements and the available experimental data on the branching ratio $R_{\mu e^-}$ for $^{48}$Ti and $^{208}$Pb as well as the expected experimental sensitivity for $^{27}$Al employed as a target in the planned at Brookhaven $\mu^--e^-$ conversion (MECO) experiment, we extract very severe constraints for the flavor violation parameters. We especially emphasize on the constraints resulting for SUSY R-parity violating parameters.