Note on a diophantine equation (Q2543209)

A method of finding all solutions of the Diophantine equation \[ (x^2 + a) (y^2 + a) = [a((y - x)/2b)^2 + b^2]^2 \] is presented. From this it is deduced that the equation \[ (x^2 + a) (y^2 + a) = (az^2 + b^2)^2 \] has an infinitude of solutions \(x,y,z\) under certain conditions satisfied by \(a\) and \(b\). The author does not know if all of the solutions of \((x^2 + 1) (y^2 + 1) = (z^2 + 1)^2\) can be obtained by his method.