### $\mathcal{D}$-bundles and integrable hierarchies

We study the geometry of $\mathcal{D}$-bundles—locally projective $\mathcal{D}$-modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of $\mathcal{D}$-bundles; in particular, we prove that the local structure of $\mathcal{D}$-bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...