An extension of the New Standard Model, by introducing a mixing of the low mass ``active neutrinos with heavy ones, or by any model with lepton flavor violation, is considered. This leads to non-orthogonal neutrino production and detection states and to modifications of neutrino oscillations in both, vacuum and matter. The possibility of the discovery of such effects in current and future neutrino oscillation experiments is discussed. First order approximation formulas for the flavor transition probabilities in constant density matter, for all experimentally available channels, are given. Numerical calculations of flavor transition probabilities for two sets of New Physics parameters describing a single ``effective heavy neutrino state, both satisfying present experimental constraints, have been performed. Two energy ranges and several baselines, assuming both the current ($pm2sigma$) and the expected in future ($pm3%$) errors of the neutrino oscillation parameters are considered, keeping their present central values. It appears that the biggest potential of the discovery of the possible presence of any New Physics is pronounced in oscillation channels in which $ u_{e}$, $ u_{bar{e}}$ are not involved at all, especially for two baselines, $L=3000 km$ and $L=7500 km$, which for other reasons are also called ``magic for future $Neutrino Factory$ experiments.
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