Solving constrained Pell equations

Consider the system of Diophantine equations x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases P (x, y) = cy2 + d and P (x, y) = cx + d, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.