Markoff–Rosenberger triples and generalized Lucas sequences

We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ a x 2 + b y 2 + c z 2 = d x y z with $$(x,y,z)=(U_i,U_j,U_k)$$ ( x , y , z ) = ( U i , U j , U k ) , where $$U_i$$ U i denotes the i-th generalized Lucas number of first/second kind. We provide an upper bound for the minimum of the indices and we apply the result to completely resolve concrete equations, e.g. we determine solutions containing only balancing numbers and Jacobsthal numbers, respectively.