The symplectic leaves of the affine Poisson $C^{\mathrm{\infty }}$-scheme associated with a singular subalgebroid

Every singular subalgebroid, and in particular every singular foliation $\mathcal{G}$, determines a universal Poisson $C^{\mathrm{\infty }}$-scheme $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{F}(\mathcal{G}))$, which is the analogue of the linear Poisson structure on the dual of a Lie algebroid. We show that the underlying space $X_{\mathcal{F}(\mathcal{G})}$ of $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{F}(\mathcal{G}))$ decomposes as a disjoint union of symplectic manifolds. Furthermore, we extend the Helffer-Nourrigat cone to a singular subalgebroid and show that it is an embedded Poisson $C^{\mathrm{\infty }}$-subscheme of $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{F}(\mathcal{G}))$.