  We study the semigroup $\textbf{{ID}}_{\infty}$ of all partial isometries of
the set of integers $\mathbb{Z}$. It is proved that the quotient semigroup
$\textbf{{ID}}_{\infty}/\mathfrak{C}_{\textsf{mg}}$, where
$\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence, is isomorphic to
the group ${\textsf{Iso}}(\mathbb{Z})$ of all isometries of $\mathbb{Z}$,
$\textbf{{ID}}_{\infty}$ is an $F$-inverse semigroup, and
$\textbf{{ID}}_{\infty}$ is isomorphic to the semidirect product
${\textsf{Iso}}(\mathbb{Z})\ltimes_\mathfrak{h}\mathscr{P}_{\!\infty}(\mathbb{Z})$
of the free semilattice with unit $(\mathscr{P}_{\!\infty}(\mathbb{Z}),\cup)$
by the group ${\textsf{Iso}}(\mathbb{Z})$. We give the sufficient conditions on
a shift-continuous topology $\tau$ on $\textbf{{ID}}_{\infty}$ when $\tau$ is
discrete. A non-discrete Hausdorff semigroup topology on
$\textbf{{ID}}_{\infty}$ is constructed. Also, the problem of an embedding of
the discrete semigroup $\textbf{{ID}}_{\infty}$ into Hausdorff compact-like
topological semigroups is studied.
