  Symplectic symmetry approach to clustering (SSAC) in atomic nuclei, recently
proposed, is modified and further developed in more detail. It is firstly
applied to the light two-cluster $^{20}$Ne + $\alpha$ system of $^{24}$Mg, the
latter exhibiting well developed low-energy $K^{\pi} = 0^{+}_{1}$, $K^{\pi} =
2^{+}_{1}$ and $K^{\pi} = 0^{-}_{1}$ rotational bands in its spectrum. A simple
algebraic Hamiltonian, consisting of dynamical symmetry, residual and vertical
mixing parts is used to describe these three lowest rotational bands of
positive and negative parity in $^{24}$Mg. A good description of the excitation
energies is obtain by considering only the $SU(3)$ cluster states restricted to
the stretched many-particle Hilbert subspace, built on the leading Pauli
allowed $SU(3)$ multiplet for the positive- and negative-parity states,
respectively. The coupling to the higher cluster-model configurations allows to
describe the known low-lying experimentally observed $B(E2)$ transition
probabilities within and between the cluster states of the three bands under
consideration without the use of an effective charge.
