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 i n the system are either single equations, or pairs of equations (not couple d with the rest) which constitute the real and imaginary part of a single c omplex equation. The theory originates from a natural extension of earlier results by allowing the Jacobian endomorphism of the system, which is assum ed to be diagonalizable, to have both real and complex eigenvalues. An impo rtant 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.