SE-SKOLORNAS 1981 Problem 2

Does $\left\{ \begin{array}{l} x^y = z y^z = x z^x = y \end{array} \right.$ have any solutions in positive reals apart from $x = y = z= 1$ ?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.