Looking for Carroll particles in two time spacetime

We make an attempt to describe Carroll particles with a nonvanishing value of energy (i.e. the Carroll particles which always stay in rest) in the framework of two time physics, developed in the series of papers by I. Bars and his coauthors. In the spacetime with one additional time dimension and one additional space dimension one can localize the symmetry which exists between generalized coordinate and their conjugate momenta. Such a localization implies the introduction of the gauge fields, which in turn implies the appearance of some first-class constraints. Choosing different gauge-fixing conditions and solving the constraints one obtain different time parameters, Hamiltonians, and generally, physical systems in the standard one time spacetime. In this way such systems as nonrelativistic particle, relativistic particles, hydrogen atoms, and harmonic oscillators were described as dual systems in the framework of two time physics. Here, we find a set of gauge fixing conditions which provides as with such a parametrization of the phase-space variables in the two time world which gives the description of Carroll particle in the one time world. Besides, we construct the quantum theory of such a particle using an unexpected correspondence between our parametrization and that obtained by Bars for the hydrogen atom in 1999.