Complexity of ANOTHER HAMILTONIAN CIRCUIT problem
This result is proved in the senior thesis The complexities of puzzles, cross sum and their another solution problems (ASP) by Takahiro Seta.

The systematic study of ASPs was initiated by Ueda and Nagao in their paper NP-completeness Results for NONOGRAM via
Parsimonious Reductions. See also Takayuki Yato's master thesis, Complexity and completeness of finding another solution and its application to puzzles.