System of equations: x – 1/x = y, y – 1/y = z, z – 1/z = x

Determine whether or not the following system has any integer solutions

$x \; - \; 1/x \; = \; y$
$y \; - \; 1/y \; = \; z$
$z \; - \; 1/z \; = \; x$