In this paper, we present a detailed study of the oscillating magnetic relaxation in the synthetic antiferromagnet (SAF) with two ferromagnetic Co layers of different thicknesses separated by an Ir spacer. The four stable magnetic states of the SAF are determined by the mutual alignment of magnetic moments in the layers and are controlled by both the magnetic interlayer exchange interaction and the Zeeman energy. The specific variations in the thicknesses of the layers and/or temperature allow the existence of a ``triple point,'' which corresponds to a coincidence of the critical switching fields for two or three interstate transitions. In this case, two or even three different types of magnetization reversals occur simultaneously and competitively. A nonmonotonic dependence of the domain-wall speed ${v}_{\mathrm{DW}}$ on magnetic field $H$ and an oscillating time dependence of magnetic moment $M$ in a constant magnetic field were observed in a Pt/Co/Ir/Co/Pt synthetic antiferromagnet with perpendicular anisotropy due to interplay between the magnetic nuclei produced by Dzyaloshinskii-Moriya interaction. The proximity of two or three (triple-point) critical fields of SAF switching is the necessary condition for both a nonmonotonic magnetic relaxation and the oscillating time variations of the magnetic moment. The dynamical model describing the interaction and subsequent evolution of the magnetic nuclei demonstrates that this nontrivial magnetic relaxation obeys a simple Schr\odinger equation.