  The properties of a nonlinear oscillator with an additional term $k_g/x^2$,
characterizing the isotonic oscillator, are studied. The nonlinearity affects
to both the kinetic term and the potential and combines two nonlinearities
associated to two parameters, $\kappa$ and $k_g$, in such a way that for
$\kappa=0$ all the characteristics of of the standard isotonic system are
recovered. The first part is devoted to the classical system and the second
part to the quantum system. This is a problem of quantization of a system with
position-dependent mass of the form $m(x)=1/(1 - {\kappa} x^2)$, with a
$\kappa$-dependent non-polynomial rational potential and with an additional
isotonic term. The Schr\"odinger equation is exactly solved and the
$(\kappa,k_g)$-dependent wave functions and bound state energies are explicitly
obtained for both $\kappa<0$ and $\kappa>0$.
