About The Indefinite Equation X~2-kxy+y~2-lx=0

Let κ be a positive integer, A.Marlewski proved has infinite positive integer solutions (x, y) if and only if k= 3 in 2004.In 2010, P.Yuan and Y.Hu proved that the equation 4 has infinite positive solutions (x, y) if and only if (k,l)=(3,1),(3,2),(3,4),(4,2),(4,4),(6,4).In 2013, L.Feng proved that if l> 0, there are only finite k can make have infinite positive integer solutions (x, y). With the help of the computer, she also gave all the k whenIn this paper I focus my attention on the (k,l) which make x2 -kxy+y2 - lx = 0 have infinite positive integer solutions. And do a research on the infinite positive integer solutions of the other two equations prove them with the elementary methods.The main conclusions:Theorem 2.1:Let k,l ∈ N*,if and only if the equation has infinite positive integer solutions (x,y).Theorem 2.2: 1(mod 4),pi are odd prime numbers, the diophantine equation has infinite positive integer solutions (x, y).Theorem 2.3:Let k ∈ N*, if and only-1 has positive integer solution, the equation has infinite positive integer solutions (x, y).