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

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