I'm trying to find positive integer solutions to the ellipse $$x^2 - xy + y^2 - k^2 = 0$$ where $k$ is a constant. Specifically, I already have two solutions for a given $k$, and I'm trying to find a third, possibly by using the two known solutions. (I'm trying to copy the technique of producing rational solutions to a conic from one known rational point on the conic.)
I have searched a lot on the Internet, but most resources either suggest checking all values within the range of the ellipse's 'box', or give methods for a particular type of ellipse. Edit: I found some questions which are similar to mine, but I cannot apply the technique used in them to my question mainly because I do not understand the technique. They all mention work by Fricke and Klein (1897).
My questions are:
- How many [positive] integer solutions does a general ellipse have?
- How can we find them? (From scratch, or knowing a few solutions beforehand?)