Anyone who likes $K3$ surfaces cares about lattices of the form $$ (2d)\cdot y^2 - 2x \cdot z$$ (namely the mukai pairing on $H^*_{alg}(K3)$ of picard $1$ with polarization $d$).

Inside we have special vectors, those $\delta$ with $\delta^2 =-2$, called roots.

We can reflect along roots.

Note the equation $$(2d)\cdot y^2 - 2\cdot x \cdot z = -2$$ has many solutions; any factorization of $(2d)y^2 + 2$ gives one.

Does someone have a reference to a study of these?

For example here is a concrete question; are the roots finitely generated by some finite subset?

The question is do we have a good understanding of the structure of the solutions via reflections? Probably the right generalization is roots in even indefinite lattices (maybe nonunimodular!).

For example had the lattice been of rank $2$ instead of $3$, we'd basically be solving for roots in a real quadratic extension; i.e by dirichlet we can generate all solutions from two of them.