I do not know if the result is trivial or known, but your result is fun and follows from the identity $$(x-1)^4+(x+1)^4+16=2(x^2+3)^2$$ so that you find infinitely many solutions by solving the Pell equation $x^2-3z^2=-3$, so $x+z\sqrt{3}=\pm\sqrt3(2+\sqrt3)^n$.
Henri Cohen
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