  The current Super-Kamiokande data on the D-N asymmetry between the the day
event rate and the {\it Night} and {\it Core} event rates, produced by solar
neutrinos which respectively cross the Earth along any trajectory and cross the
Earth core before reaching the detector, imply rather stringent constraints on
the MSW small mixing angle (SMA) $\nu_e \to \nu_{\mu(\tau)}$ solution of the
solar neutrino problem. A simplified analysis shows, in particular, that a
substantial subregion of the SMA solution region is disfavored by these data.
The {\it Core} D-N asymmetry data alone allow to rule out at 99.7% C.L. a part
of this subregion. The constraints on the MSW large mixing angle and LOW $\nu_e
\to \nu_{\mu(\tau)}$ solutions as well as on the MSW $\nu_e \to \nu_{s}$
solution, following from the data on the {\it Night} and {\it Core} D-N
asymmetries are also discussed.
