  We study the stability of the two--neutrino MSW solution of the solar
neutrino problem, corresponding to solar $\nu_e$ transitions into sterile
neutrino, $\nu_e \rightarrow \nu_s$, with respect to changes of the total
fluxes of $^{8}$B and $^{7}$Be neutrinos, $\Phi_{{\rm B}}$ and $\Phi_{{\rm
Be}}$. For any value of $\Phi_{{\rm Be}}$ from the interval $0.7\Phi^{{\rm
BP}}_{{\rm Be}}\leq \Phi_{{\rm Be}} \leq 1.3\Phi^{{\rm BP}}_{{\rm Be}}$ (for
$\Phi_{{\rm Be}} = 0.7\Phi^{{\rm BP}}_{{\rm Be}}$) the $\nu_e \rightarrow
\nu_{s}$ MSW transitions provide at 95\% C.L. a description of the existing
solar neutrino data for $0.40~(0.39)~\Phi^{{\rm BP}}_{{\rm B}} \ltap \Phi_{{\rm
B}} \ltap 36~(40)\Phi^{{\rm BP}}_{{\rm B}}$, $\Phi^{{\rm BP}}_{{\rm B}}$ and
$\Phi^{{\rm BP}}_{{\rm Be}}$ being the fluxes in the solar model of
Bahcall--Pinsonneault from 1992. The corresponding allowed regions of values of
the parameters \dm2 and \sn2, characterizing the solar neutrino transitions,
are derived. The physical implications of the found MSW $\nu_e \rightarrow
\nu_s$ solutions for the future solar neutrino experiments are considered as
well.
