A normalized family of representations of the group of motions of a Euclidean space and the inverse problem of the representation theory of this group

There exists a well-known holomorphic family $\mathscr T^\lambda$ of representations of the isometry group of $\mathbb R$ such that $\mathscr T^{-\lambda}\sim\mathscr T^\lambda$ for $\lambda\ne0$. This paper presents a holomorphic family $V_R^{\lambda}$, $|\lambda|<R$, such that $V_R^\lambda\sim\mathscr T^\lambda$ and $V_R^{-\lambda}= V_R^\lambda$ for $\lambda\ne0$. It is used for the construction of (generally speaking, reducible) representations of a fairly general form.