  We discuss a version of Hamiltonian (2+1)-dimensional dynamics, in which one
allows nonvanishing Poisson brackets also between the coordinates, and between
the momenta. The resulting equations of motion are not any more derivable from
a Lagrangian. However, taking a specific limit, in which the symplectic form
becomes singular, one can recover a first-order Lagrangian description. This
signals the dimensional reduction of the phase-space to half its initial number
of degrees of freedom. We reach the same limit from another point of view,
studying a particular form of the Poisson brackets, which is singled out
geometrically and easy to handle algebraically. For comparison, a discussion of
quantum mechanics with extended Heisenberg algebra is included. The quantum
theory constrains the antisymmetric matrix providing the algebra to the above
mentioned classically singular limit.
