  We study isometric representations of the semigroup $\mathbb{Z}_+\backslash
\{1\}$. Notion of an inverse representation is introduced and a complete
description (up to unitary equivalence) of such representations is given. Also,
we study a class of non-inverse irreducible representations --
$\beta$-representations of the semigroup $\mathbb{Z}_+\backslash \{1\}$.
