I don't know how to solve the following problem for integer solution $(x,y,z)$:
$77x^{12}-49y^{12}+2z^{12}=63xyz^{10}$
When considering this equation modulo 7 a few times, it would seem that an "infinite descent" of each variable having 7 as a divisor occurs. Does that imply that the only integer solution is trivial, that is $(0,0,0)$?
yes, because the method you used proves that the numbers are divisible by $7^n$ for any $n\in N$
by https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic, any nonzero number has unique decomposition with finitely many 7's