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

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A. de Sole, V. G. Kac, and M. Wakimoto (arXiv:1004.5387) have recently introduced a new family of compatible Hamiltonian operators of the form H⁽ⁿ,⁰⁾=D²∘((1/u)∘ D)²ⁿ∘ D, where N=2n+3, n=0,1,2,..., u is the dependent variable and D is the total derivative with respect to the independent variable. We present a differential substitution that reduces any linear combination of these operators to an operator with constant coefficients and linearizes any evolution equation which is bi-Hamiltonian with respect 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 3.