  We establish a connection between the representation theory of certain
noncommutative singular varieties and two-dimensional lattice models.
Specifically, we consider noncommutative biparametric deformations of the fiber
product of two Kleinian singularities of type $A$. Special examples are closely
related to Lie-Heisenberg algebras, the affine Lie algebra $A_1^{(1)}$, and a
finite W-algebra associated to $\mathfrak{sl}_4$.
  The algebras depend on two scalars and two polynomials that must satisfy the
Mazorchuk-Turowska Equation (MTE), which we re-interpret as a quantization of
the ice rule (local current conservation) in statistical mechanics. Solutions
to the MTE, previously classified by the author and D. Rosso, can accordingly
be expressed in terms of multisets of higher spin vertex configurations on a
twisted cylinder.
  We first reduce the problem of describing the category of weight modules to
the case of a single configuration $\mathscr{L}$. Secondly, we classify all
simple weight modules over the corresponding algebras
$\mathcal{A}(\mathscr{L})$, in terms of the connected components of the
cylinder minus $\mathscr{L}$. Lastly, we prove that $\mathcal{A}(\mathscr{L})$
are crystalline graded rings (as defined by Nauwelaerts and Van Oystaeyen), and
describe the center of $\mathcal{A}(\mathscr{L})$ explicitly in terms of
$\mathscr{L}$.
  Along the way we prove several new results about twisted generalized Weyl
algebras and their simple weight modules.
