  In this note we consider Diophantine equations of the form \begin{equation*}
a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad
\frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even
positive integers $p,q,r,s$. We show that in each case the set of rational
points on the underlying surface is dense in the Zariski topology. For the
surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the
Euclidean topology. Moreover, in this case we construct infinitely many
parametric solutions in coprime polynomials. The same result is true for
$(p,q,r,s)\in\{(2,4,8,8), (2,8,4,8)\}$. In the case $(p,q,r,s)=(4,4,4,4)$, we
present some new parametric solutions of the equation $x^4-y^4=4(z^4-w^4)$.
