AUSTRIAN-POLISH 1991 Problem 8

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \, (mod z) yz \equiv 1 (mod x) zx \equiv 1 (mod y)\end {cases}$$ Find the number of triples $(x,y, z)$ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$ .