  We consider the Whittaker modules $M_{1}(\lambda,\mu)$ for the Weyl vertex
algebra $M$, constructed in arXiv:1811.04649, where it was proved that these
modules are irreducible for each finite cyclic orbifold $M^{\Bbb Z_n}$. In this
paper, we consider the modules $M_{1}(\lambda,\mu)$ as modules for the ${\Bbb
Z}$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex
algebra $\mathcal W_{1+\infty, c=-1} = \mathcal M(2) \otimes M_1(1)$ which is
the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet
algebra $\mathcal M(2)$. Furthermore, these modules are also modules of the Lie
algebra $\widehat{\mathfrak gl}$ with central charge $c=-1$. We prove they are
reducible as $\widehat{\mathfrak gl}$-modules (and therefore also as
$M^0$-modules), and we completely describe their irreducible quotients
$L(d,\lambda,\mu)$.
  We show that $L(d,\lambda,\mu)$ in most cases are not tensor product modules
for the vertex algebra $ \mathcal M(2) \otimes M_1(1)$. Moreover, we show that
all constructed modules are typical in the sense that they are irreducible for
the Heisenberg-Virasoro vertex subalgebra of $\mathcal W_{1+\infty, c=-1}$.
