  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.
