  We study the problem of determining, given an integer $k$, the rational
solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$,
the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three
elliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine the
rational points on $C_{k}$ under the assumption that one of these elliptic
curves has rank zero. We discuss the challenges involved in extending our
result to handle all $k \in \mathbb{Q}$.
