Holomorphic automorphisms of Markov-type surfaces

Every complex surface of Markov type, i.e.\ the variety given by $x^2+y^2+z^2+Exyz-Ax-By-Cz-D=0$, has the symplectic density property and the Hamiltonian density property. We prove a singular symplectic version of the Anders{é}n–Lempert theorem for normal reduced affine complex varieties and apply it to describe the holomorphic symplectic automorphisms of a complex surface of Markov type. To this end, we also investigate the germs of vector fields in isolated singularities of type $A_k$ and $D_k$. Moreover, we show that any injective self-map of the set of ordered Markov triples can be realized by a holomorphic symplectic automorphism.