Hamiltonian Systems of Evolution Equations.

Hamiltonian operators and their behavior under differential substitutions are studied. Scalar Hamiltonian operators are classified up to fifth order, and its is shown that all such operators may be obtained from the first order Gardner operator, D_{rm x}, by differential substitutions, thus providing an infinite dimensional Darboux theorem for Hamiltonian systems of evolution equations. Compatibility conditions for bi-Hamiltonian systems of low order are found, and a strengthened version of Magri's theorem is given.