On the Diophantine equations of the form x^2-kxy+y^2+kx=0 and x^2-kxy+y^2+2kx=0.

For an integer k, define S(k) as the set of integers l where the equation x^2-lxy+y^2+kx=0 has infinitely many positive integer solutions. We demonstrate that for positive integer k, k \in S(k) if and only if k=3, 4, 5; and for k \equiv 0,1,3 mod(4), k \in S(2k) if and only if k=3, 4, 5.