Hamiltonian Cycle as an integer linear programming problem
Describing connectivity in integer programming is not as straightforward as the rest of the reduction, I think.

However, it is quite clean to reduce HAMCYCLE to SAT (e.g. here), and it is very clean to reduce SAT to integer programming. I think you will end up with a system that is not all that complicated. Perhaps it is worth doing so than busting your head over a direct reduction.