Question 384055
{{{(x-2)^2-(x-2)(y+1)+(y+1)^2=27}}}
{{{(x^2-4x+4)-(xy+x-2y-2)+(y^2+2y+1)=27}}}
{{{x^2-4x+4-xy-x+2y+2+y^2+2y+1=27}}}
{{{x^2-xy-5x+2y+y^2+4y+7=27}}}
{{{x^2-xy-5x+2y+y^2+4y-20=0}}}
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The general equation for a conic section is,
{{{Ax2 + Bxy + Cy2 + Dx + Ey + F = 0}}}
In this case,
{{{A=1}}}
{{{B=-1}}}
{{{C=1}}}
{{{D=-5}}}
{{{E=4}}}
{{{F=-20}}}
{{{B^2-4AC=1-4(1)(1)=-3}}}
So this not a parabola or a hyperbola.
It turns out to be the equation of an off axis ellipse. 
It does have several integer solutions,
(-4,-4)
(-1,-7)
(-1,2)
(5,-4)
(5,5)