Mikheyev-Smirnov-Wolfenstein effect with flavor-changing neutrino interactions

We consider the effect that flavor-nondiagonal neutrino interactions with matter have on the resonant $\nu$ oscillations. It is shown that, even in the absence of $\nu$ mixing in a vacuum, an efficient conversion of the electron neutrinos from the Sun to another $\nu$ flavor can result if the strength of this interaction is $\sim {10}^{-2\mathrm{}}{G}_F$. We show how this can be implemented in the minimal supersymmetric standard model with $R$-parity breaking. Here, the $L$-violating couplings induce neutrino masses, mixings, and the flavor-nondiagonal neutrino interactions that can provide a Mikheyev-Smirnov-Wolfenstein-like solution to the solar-neutrino problem even for negligible vacuum mixings.