  We use bounds of mixed character sums modulo a square-free integer $q$ of a
special structure to estimate the density of integer points on the hypersurface
$$ f_1(x_1) + \ldots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} $$ for some
polynomials $f_i \in {\mathbb Z}[X]$ and nonzero integers $a$ and $k_i$, $i=1,
\ldots, n$. In the case of $$ f_1(X) = \ldots = f_n(X) = X^2\quad \text{and}
\quad k_1 = \ldots = k_n =1 $$ the above hypersurface is known as the
Markoff-Hurwitz hypersurface, while for $$ f_1(X) = \ldots = f_n(X) = X^n\quad
\text{and} \quad k_1 = \ldots = k_n =1 $$ it is known as the Dwork
hypersurface. Our results are substantially stronger than those known for
general hypersurfaces.
