Integrable hierarchy for homogeneous realization of the toroidal Lie algebra ${\mathcal{L}}_{r+1}^{\mathrm{t}\mathrm{o}\mathrm{r}}({\mathfrak{s}\mathfrak{l}}_{\mathrm{\ell }})$

Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra ${\mathcal{L}}_{r+1}^{\text{tor}}({\text{fsl}}_{\mathrm{\ell }})$ via a lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent this hierarchy in the form of Lax equations, and show that it is an extension of a certain reduction of the $\mathrm{\ell }$-component KP hierarchy.