Oscillator Algebra of Chiral Oscillator

For the chiral oscillator described by a second order and degenerate Lagrangian with special Euclidean group of symmetries, we show, by cotangent bundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of oscillator algebra, the central extension of special Euclidean algebra in two dimensions. This extension, defined by symplectic two-cocycle of special Euclidean algebra, seems to be an enforcement of reduction itself rooted to Casimir function.