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.
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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.