  We consider the Diophantine equations $7x^2+y^{2n}=4z^3$ and
$x^2+7y^{2n}=4z^3$. We determine all families of solutions to these equations
for $n=2, 3$, and to the first of these equations for $n=4,5$ (only
conjecturally for the second of these equations). 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 $7 < p < 10^9$, $p\not=
13$. 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 $p$. We give a conjectural description of the
set of solutions to the Diophantine equation $x^2+7y^{2p}=4z^3$, where $p\geq
7$ is a prime.
