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We study phase-space properties of critical, parity symmetric, $N$-quDitsystems undergoing a quantum phase transition (QPT) in the thermodynamic$N\to\infty$ limit. The $D=3$ level (qutrit) Lipkin-Meshkov-Glick (LMG) modelis eventually examined as a particular example.  For this purpose, we consider U$(D)$-spin coherent states (DSCS),generalizing the standard $D=2$ atomic coherent states, to define the coherentstate representation $Q_\psi$ (Husimi function) of a symmetric $N$-quDit state$|\psi>$ in the phase space $\mathbb CP^{D-1}$ (complex projective manifold).DSCS are good variational aproximations to the ground state of a $N$-quDitsystem, specially in the $N\to\infty$ limit, where the discrete parity symmetry$\mathbb{Z}_2^{D-1}$ is spontaneously broken. For finite $N$, parity can berestored by projecting DSCS onto $2^{D-1}$ different parity invariantsubspaces, which define generalized ``Schr\"odinger cat states'' reproducingquite faithfully low-lying Hamiltonian eigenstates obtained by numericaldiagonalization. Precursors of the QPT are then visualized for finite $N$ byplotting the Husimi function of these parity projected DSCS in phase space,together with their Husimi moments and Wehrl entropy, in the neighborhood ofthe critical points. These are good localization measures and markers of theQPT.

Localization measures of parity adapted U($D$)-spin coherent states applied to the phase space analysis of the $D$-level Lipkin-Meshkov-Glick model