Complex second-order differential equations and separability

A general theory is developed about a form of maximal decoupling of systems of second-order ordinary differential equations. Such a decoupling amounts to the construction of new variables with respect to which all equations in the system are either single equations, or pairs of equations (not coupled with the rest) which constitute the real and imaginary part of a single complex equation. The theory originates from a natural extension of earlier results by allowing the Jacobian endomorphism of the system, which is assumed to be diagonalizable, to have both real and complex eigenvalues. An important tool in the analysis is the characterization of complex second-order equations on the tangent bundle TM of a manifold, in terms of properties of an integrable almost complex structure living on the base manifold M.