This suggests the full equation can be expressed as (x-2y+a)(2x+3y+b)(2x+3y+c) = abc, for some integers a,b,c.  Expanding and equating coefficients finds a=0, b=-1, c=-5 (or b/c switched).

Then (x-2y)(2x+3y-1)(2x+3y-5) = 0.  Applying the zero product property, the solutions lay on x-2y=0, 2x+3y=0, or 2x+3y-5=0.

The first line has an infinite number of positive integer solutions, parameterized as (2n,n) for some positive integer n.  The second line has no positive integer solutions.  And the third line has a single positive integer solutions at (1,1).

Then the set of positive integer solutions is (1,1) and (2n,n) where n is a positive integer.