  Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree
five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the
assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic
surface contained in $\cal{V}_{f}$. If $b=0, a<0$ and $-a\not\equiv 2,18,34
\pmod {48}$ then there exists $\Q$-rational surface contained in $\cal{V}_{f}$.
Moreover, we prove that for each $f$ of degree five there exists
$\Q(i)$-rational surface contained in $\cal{V}_{f}$.
