The Diophantine equation $p^{x}+6^{y}=z^{2n}$: a complete solution for half of the primes

Let $p$ be a prime number and let $n\geq 1$ be an integer. We completely solve the exponential Diophantine equation $p^{x}+6^{y}=z^{2n}$ in positive integers $x,y,z$ for every prime $p$ such that the Legendre symbol $\leg{6}{p}$ equals $-1$, that is, for $p\equiv 7,11,13,17\pmod{24}$: the equation h