Classification of third-order linear differential equations and symplectic sheets of the Gel'fand-Dikij bracket (Q1813400)

Linear ordinary third-order differential equations with periodic coefficients of the following form are considered: (1) \(z'''+u(x)y'+v(x)y=0\). Equation (1) is regarded as the Hamiltonian dynamical system with the symplectic structure based on the known Gel'fand-Dikij bracket. In the space of equations of the form (1), a classification of symplectic sheets of this bracket is developed. It is demonstrated that the classification problem is equivalent to finding all homotopy classes of nonflatting curves on \(S^ 2\). For equations (1) with the unit monodromy operator, there exist exactly three different sheets of the second Gelfand-Dikii bracket.