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Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by therequirement of closedness of the differential N-1 forms $\Omega_{N-1}$ of rankN-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of theseflows, given by the systems of the N-1 quasi-linear differential equations,describe coisotropic deformations of (N-1)-dimensional linear subspaces. Forthe class of solutions which are Laurent polynomials in one variable thesesystems coincide with N-dimensional integrable systems such as Liouvilleequation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3),dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4)and others. Gauge invariant part of the forms $\Omega_{N-1}$ provides us withthe compact form of the corresponding hierarchies. Dual quasi-linear systemsassociated with the projectively dual Grassmannians Gr(2,N+1) are defined viathe requirement of the closedness of the dual forms $\Omega_{N-1}^{\star}$. Itis shown that at N=3 the self-dual quasi-linear system, which is associatedwith the harmonic (closed and co-closed) form $\Omega_{2}$, coincides with theMaxwell equations for orthogonal electric and magnetic fields.

Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems