Consider the affine conic $C\subset\mathbb{A}^2_\mathbb{Z}$ cut out by $x^2 + axy + y^2 + b$, where $a,b\in\mathbb{Z}$.
Assume that $a\ne \pm 2$, and that $C$ admits an integral point $(x_0,y_0)$. The quadratic form $x^2+axy+y^2$ has discriminant $a^2-4$. Let $N := b(a^2-4)$. I believe that the base change of $C$ to $\mathbb{Z}[\frac{1}{N}]$ is a (nonsplit) algebraic torus over $\mathbb{Z}[\frac{1}{N}]$. Is this true?
Assuming this, $C(\mathbb{Z}[1/N])$ admits a group structure where $(x_0,y_0)$ is the origin. Is there a subset of $C(\mathbb{Z})$ which is also a group? Does the entirety of $C(\mathbb{Z})$ admit a group structure?
If $C$ were an elliptic curve with multiplicative reduction at every non-smooth special fiber, one obtains a group structure on the integral points which lie in the smooth locus. Is something like this still true in our case?
