On the Diophantine equation x2 − kxy + y2 + lx = 0, l ∈ {1, 2, 4}

We prove that the Diophantine equation x2−kxy+y2+lx = 0, l ∈ {1, 2, 4} has an infinite number of positive integer solutions x and y if and only if (k, l) = (3, 1), (3, 2), (4, 2), (3, 4), (4, 4), (6, 4). Furthermore, we prove that the Diophantine equation x2 −kxy+y2 + x = 0 has infinitely many integer solutions x and y if and only if k ̸= 0,±1, which answers a problem in Marlewski and Marzycki (2004) [1].