Non-symplectic symmetries and bi-Hamiltonian structures of the rational   Harmonic Oscillator
  The existence of bi-Hamiltonian structures for the rational Harmonic Oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analyzed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained. 
Jos\'e F. Cari\~nena, Giuseppe Marmo and Manuel F. Ra\~nada