On a Class of Quadratic Diophantine Equations

It is proved that the Diophantine equations $x^{2}-kxy+y^{2}+lx=0$, $l\in \{3,5\}$ have infinite number of positive integer solutions (x, y) if and only if (k,l)=(3,3),(4,3),(5,3),(3,5),(5,5),(7,5).