  Hamiltonian formulation of the string with dynamical geometry and
two-dimensional gravity with torsion is given. Canonical Hamiltonian equals to
the linear combination of first class constraints satisfying closed algebra. It
is the semidirect sum of the Virasoro algebra and the abelian subalgebra
corresponding to the local Lorentz rotation. After making the canonical
transformation the theory is quantized. It is proved that there exists Fock
space representation of pure two-dimensional gravity with torsion containing no
central charge in the Virasoro algebra. Also constructed is the new Fock
representation of a standard bosonic string. It is shown that two-dimensional
string with dynamical geometry is anomaly free and describes two physical
degrees of freedom.
