Representations of the Poincaré group in $E(2)$ bases

Constructions are given of irreducible unitary representations of the quantum-mechanical Poincaré group in two bases associated with the $E(2)$ subgroup for the ease when the square of the mass and the energy are positive, i.e., $M^2>0$ and $P_0>0$. Expressions are found for the generators and the transformation functions from a canonical basis to the $E(2)$ bases are caleuiated. The reduction of a direct product of irreducible representations in an $E(2)$ basis is considered at the end of the paper.