Conjectures on classification of Hamiltonian operators of arbitrary odd order

Let $H$ be a Hamiltonian operator in one differential variable $u$, of odd order $N$. Write $H^{(N,c)}$, $H^{(N,1)}$, and $H^{[N,c(x)]}$ for the classified families of Hamiltonian operators, let $H^{(j,0)}$ denote the corresponding operators of odd order $j$, and let $\mathcal{C}$ be the field of constants. A contact transformation is a transformation of the indicated form $x=y$, $u=v^{n+1}/(n+1)$, possibly followed by an additive constant in $u$. Two operators are equivalent when related by the equivalences used in the classification.

Classification conjectures. (1) Every Hamiltonian operator is equivalent to one whose leading coefficient is $1$. For $N=2n+3\geq 3$, this is equivalent to an operator with leading coefficient $u^{3-N}$ by the contact transformation $x=y$, $u=v^{n+1}/(n+1)$. (2) If $H$ is a translation-invariant, nonconstant-coefficient Hamiltonian operator of order $N=2n+3\geq 7$ with leading coefficient $1$, then after the contact transformation $x=y$, $u=v^{n+1}/(n+1)+\operatorname{const.}$ it becomes either $H^{(N,c)}$ with $c\in\mathcal{C}$, or a constant-coefficient linear combination of the operators $H^{(j,0)}$ for odd $j$ satisfying $3\leq j\leq N$. (3) Every translation-invariant Hamiltonian operator of order $N\geq 7$ is equivalent either to a quasiconstant-coefficient skew-adjoint differential operator or to a linear combination of the operators $H^{(j,0)}$ for odd $j$ satisfying $3\leq j\leq N$. (4) For $N\geq 7$, $H^{(N,1)}$ is compatible with no translation-invariant Hamiltonian operator other than a scalar multiple of itself. (5) Every Hamiltonian operator of order $N\geq 13$ is equivalent either to a quasiconstant-coefficient skew-adjoint differential operator, to a constant-coefficient linear combination of the operators $H^{(j,0)}$ for odd $j$ satisfying $3\leq j\leq N$, or to $H^{[N,c(x)]}$ with $c”(x)=0$. (6) For $N\geq 11$, $H^{[N,c(x)]}$ with $c'(x)\neq 0$ and $c”(x)=0$ is compatible only with a constant multiple of itself.

These claims propose a classification of Hamiltonian operators in one differential variable at arbitrary odd order, while also describing compatibility restrictions for the two nonconstant families. The paper presents them as conjectures; no resolution status is supplied in the source.