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 ${\mathcal{P}}_1$ is a local Poisson bracket of order one and ${\mathcal{P}}_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the point transformation. From the description of Casimirs we derive new Harry Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number of components. In two component case our HS equation differs from the well-known HS2 equation.