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.
Martin,A new proof of Chen ’s theorem for Markoff graphs, Inven- tiones Mathematicae, Volume 241, pages 623–626 (2025)
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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.