Finding integer solutions for general equation; $ax^2 + bxy + cy^2 + dx + ey + f = 0$

Question

I'm trying to code a program for finding integer solutions for the general hyperbola equation; $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$

I'm basing it on a paper I found on arXiv; arXiv:0907.3675 [math.GM]

In a nutshell, the paper derives an equation; $$p * q = I$$ and creates systems of linear equations based on factors of |I|, such that; $$p = a_px + b_py + c_p = \pm d_i$$ $$q = a_qx + b_qy + c_q = I / \pm d_i$$ and $$p = a_px + b_py + c_p = I / \pm d_i$$ $$q = a_qx + b_qy + c_q = \pm d_i$$

My first candidate equation is; $$x^2 - y^2 - y = -5$$ which gives; $$a=1,b=0,c=-1,d=0,e=-1,f=5$$

The program then runs;

(1)
    ax^2 + bxy + cy^2 + dx + ey + f  =  0,  
           a = 1, b = 0, c = -1, d = 0, e = -1, f = 5


k:
    k^2  =  b^2 - 4ac
         =  0^2 - 4(1)(-1)
         =  4,  k  =  2

(10)
    p * q  =  I

    p  =  2akx + k(b - k)y + dk + (2ae - bd)
       =  2(1)(2)x + 2(0 - 2)y + 0(2) + (2(1)(-1) - (0)(0))
       =  4x + -4y + -2

    q  =  2akx + k(b + k)y + dk - (2ae - bd)
       =  2(1)(2)x + 2(0 + 2)y + 0(2) - (2(1)(-1) - (0)(0))
       =  4x + 4y - -2

    I  =  k^2(d^2 - 4af) - (2ae - bd)^2
       =  2^2(0^2 - 4(1)(5)) - (2(1)(-1) - (0)(0))^2))
       =  -84

    [4x + -4y + -2] * [4x + 4y - -2]  =  -84


factor pairs of I 
    (d_i, I/d_i)  =  [(1, -84), (2, -42), (3, -28), (4, -21), (6, -14), (7, -12), 
                      (-1, 84), (-2, 42), (-3, 28), (-4, 21), (-6, 14), (-7, 12)]


Linear systems
    p  =  4x + -4y + -2  =  d_i         p  =  4x + -4y + -2  =  I / d_i
    q  =  4x + 4y - -2  =  I / d_i       q  =  4x + 4y - -2  =  d_i


    p  =  4x + -4y + -2  =  1       p  =  4x + -4y + -2  =  -84
    q  =  4x + 4y - -2  =  -84       q  =  4x + 4y - -2  =  1

    ... and so on ...

The problem is; All of my coefficients in p and q are even, but some of the factors of |I| are odd, and so it is immediately apparent that not all solutions are in fact integers.

Is this approach totally off kilter somehow, or should I just ignore the non-integer solutions?

Also, the paper I'm following was the only ready reference I found on this process. Pointers to other good, specific references would be appreciated.