  We show that it is undecidable whether a system of linear equations over the
Laurent polynomial ring $\mathbb{Z}[X^{\pm}]$ admit solutions where a specified
subset of variables take value in the set of monomials $\{X^z \mid z \in
\mathbb{Z}\}$. In particular, we construct a finitely presented
$\mathbb{Z}[X^{\pm}]$-module, where it is undecidable whether a linear equation
$X^{z_1} \boldsymbol{f}_1 + \cdots + X^{z_n} \boldsymbol{f}_n =
\boldsymbol{f}_0$ has solutions $z_1, \ldots, z_n \in \mathbb{Z}$. This
contrasts the decidability of the case $n = 1$, which can be deduced from
Noskov's Lemma.
  We apply this result to settle a number of problems in computational group
theory. We show that it is undecidable whether a system of equations has
solutions in the wreath product $\mathbb{Z} \wr \mathbb{Z}$, providing a
negative answer to an open problem of Kharlampovich, L\'{o}pez and Miasnikov
(2020). We show that there exists a finitely generated abelian-by-cyclic group
in which the problem of solving a single quadratic equation is undecidable. We
also construct a finitely generated abelian-by-cyclic group, different to that
of Mishchenko and Treier (2017), in which the Knapsack Problem is undecidable.
In contrast, we show that the problem of Coset Intersection is decidable in all
finitely generated abelian-by-cyclic groups.
