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

A Quantum Quasi-Harmonic Nonlinear Oscillator with an Isotonic Term