  We prove that the extended Poincare group in (1+1) dimensions is
non-nilpotent solvable exponential, and therefore that it belongs to type I. We
determine its first and second cohomology groups in order to work out a
classification of the two-dimensional relativistic elementary systems.
Moreover, all irreducible unitary representations of the extended Poincare
group are constructed by the orbit method. The most physically interesting
class of irreducible representations corresponds to the anomaly-free
relativistic particle in (1+1) dimensions, which cannot be fully quantized.
However, we show that the corresponding coadjoint orbit of the extended
Poincare group determines a covariant maximal polynomial quantization by
unbounded operators, which is enough to ensure that the associated quantum
dynamical problem can be consistently solved, thus providing a physical
interpretation for this particular class of representations.
