Bartz-Marlewski equation with generalized Lucas components

Let $\lbrace U_n\rbrace =\lbrace U_n(P,Q)\rbrace $ and V_n\rbrace V_n(P,Q)\rbrace be the Lucas sequences of first second kind respectively at parameters $P \ge 1$ $Q \in \lbrace -1, 1\rbrace $. In this paper, we provide a technique for characterizing solutions so-called Bartz-Marlewski equation \[ x^2-3xy+y^2+x=0\,, \] where $(x,y)=(U_i, U_j)$ or $(V_i, V_j)$ with $i$, j 1$. Then, procedure is applied to completely resolve certain values such parameters.