BALTIC-WAY 1995 Problem 1

Find all triples $(x, y, z)$ of positive integers satisfying the system of equations

$$\left\{\begin{array}{l} x^{2}=2(y+z) \\ x^{6}=y^{6}+z^{6}+31\left(y^{2}+z^{2}\right) \end{array}\right.$$