The motion of a spherical bubble in a nonlinear viscoelastic media subjected to an acoustic pressure field is considered. The ambient fluid is composed of a Newtonian liquid in which additives at small volume fraction are diluted. The contribution of the additives with high aspect ratio brings strong anisotropy and is described by an extensional viscosity. The elastic effect is presented by the relaxation time of the additives. A lower convected Maxwell model is adopted to describe the viscoelastic properties, resulting in a modified Rayleigh-Plesset equation. The set of governing equations does not require a numerical solution for the space domain. Non-linear radial oscillations of a single bubble are obtained numerically using a fifth order Runge-Kutta scheme with adaptive time step. The results predict an extra anisotropy for a Deborah number regime De similar to 1, due to stretched additives, which contributes to bubble motion stabilization. Under this condition, the relaxation time is greater than the time scale of the flow, where no interaction between the elastic effect of the additives and the motion of the bubble is found. However, for De similar to 0.1 we observe an increase of vibrational modes on the frequency domain and higher bubble internal pressure, which may lead to collapse occurrence. The decrease in the volume fraction of the additives also shows significant variation of bubble oscillations as the elastic effect has a proportionally larger contribution than the anisotropic effect. Other results and considerations regarding relevant parameters are also discussed. (C) 2012 Elsevier B.V. All rights reserved.