For most parameters with nonzero 3+a1+a2+a3 and p>=5, every nontrivial orbit on the generalized Markov surface has size divisible by p, with quadratic obstructions giving at least two or four orbits in special cases.
Theory of Groups of Finite Order
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Divisibility by $p$ for Markoff-like Surfaces
For most parameters with nonzero 3+a1+a2+a3 and p>=5, every nontrivial orbit on the generalized Markov surface has size divisible by p, with quadratic obstructions giving at least two or four orbits in special cases.