Geometry of Inhomogeneous Poisson Brackets, Multicomponent Harry Dym Hierarchies, and Multicomponent Hunter–Saxton Equations

We introduce a natural class of multicomponent local Poisson structures $$\mathcal P_k + \mathcal P_1$$ , where is bracket order one and P_k$$ homogeneous odd $$k$$ under the assumption that has Darboux coordinates (Darboux–Poisson bracket) nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), provide description Casimirs, using purely algebraic procedure. In two-component case, completely classify brackets up point transformation. From derive new Harry Dym (HD) hierarchies Hunter–Saxton (HS) equations for number components. our HS equation differs from well-known HS2 equation. DOI 10.1134/S1061920822040100