  We consider the Markoff-Rosenberger equation $$ax^2+by^2+cz^2=dxyz$$ with
$(x,y,z)=(U_i,U_j,U_k),$ where $U_i$ denotes the $i$-th generalized Lucas
number of first/second kind. We provide 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.
