  We consider the Diophantine equation $7x^{2} + y^{2n} = 4z^{3}$. We determine
all solutions to this equation for $n = 2, 3, 4$ and $5$. We formulate a Kraus
type criterion for showing that the Diophantine equation $7x^{2} + y^{2p} =
4z^{3}$ has no non-trivial proper integer solutions for specific primes $p >
7$. We computationally verify the criterion for all primes $7 < p < 10^9$, $p
\neq 13$. We use the symplectic method and quadratic reciprocity to show that
the Diophantine equation $7x^{2} + y^{2p} = 4z^{3}$ has no non-trivial proper
solutions for a positive proportion of primes $p$. In the paper \cite{ChDS} we
consider the Diophantine equation $x^{2} +7y^{2n} = 4z^{3}$, determining all
families of solutions for $n=2$ and $3$, as well as giving a (mostly)
conjectural description of the solutions for $n=4$ and primes $n \geq 5$.
