  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.
