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The regime of undamped oscillations characterizing the stellar dynamics ofvirialised systems is analysed within the framework of a new approach to thestudy of the integrals of motion. The method is relevant as far as thecosmological implementation is concerned, as it applies to the calculations aptfor any age of the evolution of the Universe starting from the epoch ofnon-Gaussianities until present time. The new method here developed is based onthe asymptotic small-parameters expansion of the new expression of theErmakov-Lewis-Leach integrals of motion; as a result, an infinite series of newconservation laws implies the uniqueness and existence of new integrals ofmotion; analytical examples are provided after the pulsed Plummer potential,the three instances of the pulsed Dehnen potentials, the pulsed harmonicpotential, the Jaffe potential, the Hernquist potential, applied to the studiedproblem. Particular cases of complex potentials are therefore also comprehendedin the analysis. The constants of motions descending from the conservation lawsare demonstrated to depend on the virialised radius and on the virialised massof the stellar system, independently of the potential used. Cosmologicalimplementation is given within the framework of a generic Terzic'-Kandruppotential. Comparison with data analysis techniques are provided with.

Stellar dynamics within the virial theorem: asymptotic small-parameters  time expansion of the Ermakov-Lewis-Leach invariant as an infinite series of  conservation laws