For nondegenerate parameters, the Vieta-involution group is transitive on all but small orbits of F_p-solutions for almost all primes; degenerate parameters yield exactly two or four large orbits.
Goldman,Ergodic theory on moduli spaces, Annals of Mathematics146(1997), no
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Strong Approximation for the Relative Character Variety of the Four-Times Punctured Sphere
For nondegenerate parameters, the Vieta-involution group is transitive on all but small orbits of F_p-solutions for almost all primes; degenerate parameters yield exactly two or four large orbits.