  Exact symmetry and symmetry-breaking phenomena play a key role in gaining a
better understanding of the physics of many-particle systems, from quarks and
atomic nuclei, through molecules and galaxies. In nuclei, exact and dominant
symmetries such as rotational invariance, parity, and charge independence have
been clearly established. Beyond such symmetries, the nature of nuclear
dynamics appears to exhibit a high degree of complexity, and only now, we show
the fundamental role of an emergent approximate symmetry in nuclei, the
symplectic Sp(3,R) symmetry, as clearly unveiled from ab initio studies that
start from realistic interactions. In this article, we detail and enhance our
recent findings presented in Physical Review Letters 124 (2020) 042501, that
establish Sp(3,R) as a remarkably good symmetry of the strong interaction, and
point to the predominance of a few equilibrium nuclear shapes (deformed or not)
with associated vibrations and rotations that preserve the symplectic Sp(3,R)
symmetry. Specifically, we find that the structure of nuclei below the calcium
region in their ground state, as well as in their low-lying excited states and
giant resonances, respects this symmetry at the 60-80% level.
