Quadratic diophantine equations and geometry of numbers

Let (for concreteness) $ a = 2$ , $ b = \sqrt{5}$ and $ \varphi = (\sqrt{5}+1)/2$ . I am interested in solutions $ (w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system

$ $ w^2 – ax^2 -by^2 + abz^2 = 1 $ $ $ $ \lvert w^2 + ax^2 +by^2 + abz^2 \rvert \ll \infty $ $ $ $ \lvert \bar{w}^2 + a\bar{x}^2 -b\bar{y}^2 – ab\bar{z}^2 \rvert \le C $ $ for some constant $ C$ . Here $ \overline{\alpha + \beta\sqrt{5}} = \alpha – \beta\sqrt{5}$ and “$ \ll \infty$ ” means that ideally I would like to enumerate solutions in increasing order of this value. (Restriction of scalars turns this problem into a system of two quadratic equations and two inequalities in eight variables in $ \mathbb{Z}$ ; if someone wants to see it, I can write it out including potential mistakes).

  1. What is the best (or even any practical) way to produce these?

I am aware that there is a lot of classical mathematics associated to this question but I don’t quite manage to put it together. Perhaps a subquestion is:

  1. Can one enumerate the squares $ s$ in $ \mathbb{Z}[\varphi]$ with $ \lvert \bar{s} \rvert \le C$ in increasing order of $ \lvert s \rvert$ ?

Context: Let $ k = \mathbb{Q}(\varphi)$ and let $ A$ be the quaternion algebra $ (\frac{a,b}{k})$ with norm $ \nu$ . With the above values the algebra $ A$ is a skew field but tensoring with $ \mathbb{R}$ in the two possible ways (taking $ \sqrt{5}$ to $ \pm\sqrt{5}$ ) gives an isomorphism with $ M_2(\mathbb{R})$ which we equip with the map $ $ \left\lVert\left(\begin{array}{cc}\alpha&\beta\\gamma&\delta\end{array}\right)\right\rVert = \frac{1}{2}\left(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\right). $ $ The above system then asks for solutions $ \lambda$ in the maximal order of $ A$ for $ \nu(\lambda) = 1$ , $ \lVert\lambda\rVert_{\sqrt{5} \mapsto \sqrt{5}} \ll \infty$ and $ \lVert \lambda \rVert_{\sqrt{5} \mapsto -\sqrt{5}} \le C$ .