Let $Q$ be an indefinite binary quadratic form with integer coefficients. Suppose it is primitive and irreducible over $\mathbb{Q}$. I am looking for the following fact. Let $n > 1$. Suppose $n = F(x_1, y_1)$ and $n = F(x_2, y_2)$ where $(x_1, y_1)$ and $(x_2, y_2)$ are distinct integral pairs.
I am looking for a reference for the following fact (which I think is true): Then there exists an integral automorph $T$ such that $(x_2, y_2) = T(x_1, y_1)$.
Thank you!