  Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas
sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In
this paper, we explain how to find all the solutions of the Diophantine
equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le
m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$ are given integers with
$A\ne 0,~B\ne 0$, $m,n,m_1,n_1$ are nonnegative integer unknowns and $r$ is
also unknown.
