ar X iv : m at h / 99 07 16 9 v 1 [ m at h . Q A ] 2 6 Ju l 1 99 9 Integrable deformations of Hamiltonian systems and q - symmetries 1

The complete integrability of the hyperbolic Gaudin Hamiltonian and other related integrable systems is shown to be easily derived by taking into account their sl(2, R) coalgebra symmetry. By using the properties induced by such a coalgebra structure, it can be proven that the introduction of any quantum deformation of the sl(2, R) algebra will provide an integrable deformation for such systems. In particular, the Gaudin Hamiltonian arising from the non-standard quantum deformation of the sl(2, R) Poisson algebra is presented, including the explicit expressions for its integrals of motion. A completely integrable system of nonlinearly coupled oscillators derived from this deformation is also introduced.