I have an ellipse in the form of $$x^2 + xy + y^2 = 1009,$$ which I have parameterized into the following:
$$x = \frac{-35t^2 - 16t + 27}{t^2+t+1}$$ and
$$y = \frac{27t^2+70t+8}{t^2+t+1}.$$
I saw that for the ellipse $x^2 - xy + y^2 - k^2 = 0$, integer solutions could be found through parameterization on an older forum post, which is why I tried to do the same with this ellipse. Is it possible to do this through the parameterization of this ellipse? What are the next steps after I have parameterized the ellipse?
If not, what is the widely-accepted method of solving ellipses in the integers?