  The regime of undamped oscillations characterizing the stellar dynamics of
virialised systems is analysed within the framework of a new approach to the
study of the integrals of motion. The method is relevant as far as the
cosmological implementation is concerned, as it applies to the calculations apt
for any age of the evolution of the Universe starting from the epoch of
non-Gaussianities until present time. The new method here developed is based on
the asymptotic small-parameters expansion of the new expression of the
Ermakov-Lewis-Leach integrals of motion; as a result, an infinite series of new
conservation laws implies the uniqueness and existence of new integrals of
motion; analytical examples are provided after the pulsed Plummer potential,
the three instances of the pulsed Dehnen potentials, the pulsed harmonic
potential, the Jaffe potential, the Hernquist potential, applied to the studied
problem. Particular cases of complex potentials are therefore also comprehended
in the analysis. The constants of motions descending from the conservation laws
are demonstrated to depend on the virialised radius and on the virialised mass
of the stellar system, independently of the potential used. Cosmological
implementation is given within the framework of a generic Terzic'-Kandrup
potential. Comparison with data analysis techniques are provided with.
