  We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 -
bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure
which, for any fixed $b$, either confirms that the simultaneous Pell equations
have at most one solution in positive integers, or finds all exceptions for
which we have proved that there are at most finitely many.
