  Relativistic Quantum Mechanics suffers from structural problems which are
traced back to the lack of a position operator $\hat{x}$, satisfying
$[\hat{x},\hat{p}]=i\hbar\hat{1}$ with the ordinary momentum operator
$\hat{p}$, in the basic symmetry group -- the Poincar\'e group. In this paper
we provide a finite-dimensional extension of the Poincar\'e group containing
only one more (in 1+1D) generator $\hat{\pi}$, satisfying the commutation
relation $[\hat{k},\hat{\pi}]=i\hbar\hat{1}$ with the ordinary boost generator
$\hat{k}$. The unitary irreducible representations are calculated and the
carrier space proves to be the set of Shapiro's wave functions. The generalized
equations of motion constitute a simple example of exactly solvable
finite-difference set of equations associated with infinite-order polarization
equations.
