Are the eigensolutions of a 1-d.o.f. system with viscoelastic damping oscillatory or not?

In this note, one considers a 1 d.-o.-f. oscillator consisting of a mass and a viscoelastic spring, the rheology of which is represented by a so-called Zener 3-parameters model. Such a system has three eigenmotions: two of them are damped with or without oscillations (as in the case of viscous damping), the third is damped without oscillation. Introducing appropriate parameters, one has performed a detailed discussion of the nature of these eigenmotions.