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A. de Sole, V. G. Kac, and M. Wakimoto (arXiv:1004.5387) have recentlyintroduced a new family of compatible Hamiltonian operators of the form$H^{(N,0)}=D^2\circ((1/u)\circ D)^{2n}\circ D$, where $N=2n+3$, $n=0,1,2,...$,$u$ is the dependent variable and $D$ is the total derivative with respect tothe independent variable. We present a differential substitution that reducesany linear combination of these operators to an operator with constantcoefficients and linearizes any evolution equation which is bi-Hamiltonian withrespect to a pair of any nontrivial linear combinations of the operators$H^{(N,0)}$. We also give the Darboux coordinates for $H^{(N,0)}$ for any odd$N\geqslant 3$.

The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations