  A generalized family of scalar structured Gaussian modes including
helical-Ince--Gaussian (HIG) and Hermite--Laguerre--Gaussian (HLG) beams is
presented with physical insight upon a hybrid topological evolution nature of
multi-singularity vortex beams carrying orbital angular momentum (OAM).
Considering the physical origins of intrinsic coordinates aberration and the
Gouy phase shift, a closed-form expression is derived to characterize the
general modes in astigmatic optical systems. Moreover, a graphical
representation, Singularities Hybrid Evolution Nature (SHEN) sphere, is
proposed to visualize the topological evolution of the multi-singularity beams,
accommodating HLG, HIG and other typical subfamilies as characteristic curves
on the sphere surface. The salient properties of SHEN sphere for describing the
precise singularities splitting phenomena, exotic structured light fields, and
Gouy phase shift are illustrated with adequate experimental verifications.
