Transverse sections for the second Hamiltonian KdV structure (Q1127886)

We study regular and singular operators for the second KdV Hamiltonian structure. We prove that at some point of any regulary symplectic leaf, one can move transversally to the leaf by simply adding a constant to the coefficients of the operator. We also prove that given any leaf, one can move transversally to the leaf at some point by adding certain trigonometric polynomials to the coefficient of the operator. We discuss the implications of this result for the Poisson geometry of Adler-Gel'fand-Dikii manifolds and for the normal forms of scalar operators with periodic coefficients.