Tensorial criterion for flat Kronecker bihamiltonian structures

Let a homogeneous bihamiltonian structure have type $\left(2k_{1}-1,2k_{2}-1,\dots,2k_{l}-1\right)$. Tensorial criterion for Kronecker structures. There exist $N>0$ and natural tensor fields $K_{1},\dots,K_{N}$ associated with the structure such that the structure is Kronecker if and only if

$$K_i=0,\qquad 1\leq i\leq N.$$

The claim proposes intrinsic tensorial obstructions to the Kronecker condition for homogeneous bihamiltonian structures. The source gives no resolution.