Markoff–Rosenberger triples in arithmetic progression

We study the solutions of the Rosenberg–Markoff equation ax2+by2+cz2=dxyz (a generalization of the well-known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x2+y2+z2=dxyz over quadratic fields and the classic Markoff equation x2+y2+z2=3xyz over an arbitrary number field.