Let $A$ be a matrix of some $(v,k,\lambda)$-design.
If there is a $(v,k,\lambda)$ - design and $n = k -\lambda$ is odd then the equation $ z^2=nx^2+\left(-1\right)^{\frac{v-1}{2}}\lambda y$ has nontrivial solution over $\mathbb{Z}$ (Bruck–Ryser–Chowla theorem)
Let $A_{\alpha,\beta}=\alpha A + \beta I$, $\alpha,\beta \in \mathbb{Q}$. I'm trying to find some necessary conditions of existence of $(v,k,\lambda)$ - design (as Bruck–Ryser–Chowla did) in terms of $A_{\alpha,\beta}$ - matrix.
(Note: I = $(1)_{v\times v}$ and E - identity matrix)
It's easy to see that $$\left(\alpha A + \beta I\right)^T \left(\alpha A + \beta I\right) = \left(\alpha^2\lambda + 2\alpha\beta k + \beta^2 v\right)I + \alpha^2\left(k-\lambda\right)E$$ then we get (as in Bruck–Ryser–Chowla theorem) an equation $$z^2=x^2+\left(-1\right)^{\frac{v-1}{2}}\left(\alpha^2\lambda + 2\alpha \beta k + \beta^2 v \right)y^2$$
The question: can we say something about solvability over $\mathbb{Z}$ of the last equation?