**Setup:** Let $k$ be a field, let $n$ be a positive integer, and let $R := k[[x_1,\ldots,x_n]]$ denote the commutative ring of formal power series over $k$ in $x_1,\ldots,x_n$. We know that there is exactly one maximal ideal for $R$, namely $\langle x_1,\ldots,x_n \rangle$.

By localizing at the multiples of the $x_1,\ldots,x_n$, we can construct a multivariable Laurent series ring $L$. In particular, $L$ is equal to the ring of series of the form $$\sum _{m_1,\ldots,m_n \in \mathbb{Z}} \quad \lambda _{(m_1,\ldots,m_n)} \; x^{m_1}\cdots x^{m_n},$$ for $\lambda_{(m_1,\ldots,m_n)} \in k$, but with $\lambda _{(m_1,\ldots,m_n)} = 0$ when the minimum of the $m_1,\ldots,m_n$ is $\ll 0$.

When $n = 1$ it is well known that $L$ is a field. However, for $n > 1$ the situation is more subtle. For example, when $n=2$, the ideals of the form $\langle x_1 - \mu x_2 \rangle$, for nonzero $\mu \in k$, are maximal ideals.

**My general question:** Does anyone know of an explicit description of the maximal ideals of $L$, for $n > 1$?

**A more refined question:** Suppose $k$ is algebraically closed. Consider the Laurent polynomial ring $P := k[x_1^{\pm 1},\ldots,x_n^{\pm 1}]$, and let $H$ denote the multiplicative $n$-torus $(k^\times)^n$.

There is a natural action of $H$ on $P$, obtained via $$(\alpha_1,\ldots,\alpha_n)\cdot x_i = \alpha_ix_i,$$ for $(\alpha_1,\ldots,\alpha_n) \in H$. The action of $H$ on $P$ induces an action of $H$ on the maximal spectrum of $P$, and it follows from Hilbert's Nullstellensatz that there is exactly one $H$-orbit of maximal ideals.

So the refined question is: Consider the analogous action of $H$ on $L$. Are there finitely or infinitely many $H$-orbits of maximal ideals of $L$?

**My own motivation:** These commutative Laurent series rings show up as central subalgebras of $q$-commutative Laurent series rings (i.e., formal Laurent series where $x_i x_j = q_{ij}x_j x_i$ for suitable scalars $q_{ij}$). In joint work with Linhong Wang, I have been studying $q$-commutative power and Laurent series rings.
Properties of the prime and primitive ideals in these noncommutative algebras are strongly influenced by the behavior of these central subalgebras.

Thank you for your time! Any hints or references greatly appreciated.