Solving constrained Pell equations

Consider the system of Diophantine equations $x^2 - ay^2 = b$, $P(x,y) = z^{2}$, 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 $P(x, y) = cy^2 + d$ and $P(x, y) = cx + d$, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.