  Nuclear reactions in stars occur between nuclei in the high-energy tail of
the energy distribution and are sensitive to possible deviations from the
standard equilibrium thermal-energy distribution, the well-known
Maxwell-Boltzmann Distribution (\textsf{MBD}). In a previous paper published in
Physics Letters 441B(1998)291, Degl'Innocenti {\it et al}. made strong
constrains on such deviations with the detailed helioseismic information of the
solar structure. With a small deviation parameterized with a factor
exp$[{-\delta (E/kT)^2}]$, it was shown $\delta$ restricted between -0.005 and
+0.002. These constrains have been carefully reexamined in the present work. We
find that a normalization factor was missed in the previous modified
\textsf{MBD}. In this work, the normalization factor $c$ is calculated as a
function of $\delta$. It shows the factor $c$ is almost unity within the range
0$< \delta \leq$0.002, which supports the previous conclusion. However, it
demonstrates that $\delta$ cannot take a negative value from the normalization
point of view. As a result, a stronger constraint on $\delta$ is defined as
0$\leq \delta \leq$0.002. The astrophysical implication on the solar neutrino
fluxes is simply discussed based on a positive $\delta$ value of 0.003. The
reduction of the $^7$Be and $^8$B neutrino fluxes expected from the modified
\textsf{MBD} can possibly shed alternative light on the solar neutrino problem.
In addition, the resonant reaction rates for the $^{14}$N($p$,$\gamma$)$^{15}$O
reaction are calculated with a standard \textsf{MBD} and a modified
\textsf{MBD}, respectively. It shows that the rates are quite sensitive even to
a very small $\delta$. This work demonstrates the importance and necessity of
experimental verification or test of the well-known \textsf{MBD} at high
temperatures.
