For most parameters, the group Γ acts transitively on the bulk of solutions to X² + Y² + Z² = XYZ + A X + B Y + C Z + D modulo p for density one primes p, with small orbits coming from finite complex orbits.
Sarnak,Affine sieve lecture slides, 2010, http://publications.ias.edu/sarnak/ paper/508 3
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Strong Approximation for the Character Variety of the Four-Times Punctured Sphere
For most parameters, the group Γ acts transitively on the bulk of solutions to X² + Y² + Z² = XYZ + A X + B Y + C Z + D modulo p for density one primes p, with small orbits coming from finite complex orbits.