Simple algorithm for finding a second Hamilton cycle

A classical theorem of C. A. B. Smith states that for every edge $e$ of a cubic graph $G$, the number of Hamilton cycles containing $e$ in $G$ is an even number. Tutte proved Smith's theorem using a nonconstructive parity argument. Thomason later invented the lollipop algorithm and provided a first constructive proof. We describe a simple algorithm based on Tutte's proof, thus providing an alternative constructive proof of Smith's theorem. Also this algorithm is exponential in the worst case.