  The subject matter of this work is integral points on conics described by the
general equation, ax^2+bxy+cy^2+dx+ey+f=0 (1) where the six coefficients are
integers satisfying the conditions, b^2-4ac=k^2, with a and c being nonzero and
k a positive integer. It is well known the when b^2-4ac>0, equation (1)
describes either a hyperbola on the plane or a pair of two straight lines(the
degenerate case). The key integer is the number, I=k^2(d^2-4af)-(2ae-bd)^2. In
Section 2, we show via a straightforward algebraic method that equation (1) can
be put in the form, g(x,y)h(x,y)=I, where g(x,y) and h(x,y) are linear
polynomials in x and y with integer coefficients. Thus, when I is not zero,
equation (1) has only finitely many integer solutions (x,y). The process of
finding these solutions is outlined in Section 3. In Section 4,we give a
detailed numerical example. In Section 5, we offer some observations and
remarks. In Section 6, we discuss the case b=d=e=0. In Section 7 we discuss the
case a=1=k, and we show that when I=2^n, n>or=2, the above hyperbola has
exactly 2(n-1) integral points, all explicitly given. In Section 10, we prove
that these points are distinct. In Section 8 we discuss the case I=0, the
hyperbola in (1) has either infinitely many integral points: or no integral
points. Finally in Section 9, we discuss the special case d=e=f=0.
