Triangle surfaces are exactly the Markov-type cubics xyz=x^2+y^2+z^2+ax+by+cz+d, and their automorphism groups are Gσ⋊Γ with Γ one of five finite groups from Table 1.
Diophantine equations via cluster transformations.Journal of Algebra, 462:320– 337, 2016
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Open surfaces with a triangle at infinity
Triangle surfaces are exactly the Markov-type cubics xyz=x^2+y^2+z^2+ax+by+cz+d, and their automorphism groups are Gσ⋊Γ with Γ one of five finite groups from Table 1.