  The Legendre transform expresses dynamics of a classical system through
first-order Hamiltonian equations. We consider coherent state transforms with a
similar effect in quantum mechanics: they reduce certain quantum Hamiltonians
to first-order partial differential operators. Therefore, the respective
dynamics can be explicitly solved through a flow of points in extensions of the
phase space. This generalises the geometric dynamics of a harmonic oscillator
in the Fock space. We describe all Hamiltonians which are geometrised (in the
above sense) by Gaussian and Airy beams and write down explicit solutions for
such systems.
