Lower-dimensional tori in reversible systems.

On a (2n+d)-dimensional manifold M consider a vector field V reversible with respect to an involution G whose fixed point manifold is of dimension n+d. It is conjectured that generically for each 0</=m</=n, the phase space M contains (m+d)-parameter Cantor families of m-tori invariant under both the involution G and the flow of V. To be more precise, vector fields V with this property constitute an open set in the space of all vector fields equipped with an appropriate topology. The flow of V induces on these tori quasiperiodic motions with strongly incommensurable frequencies. Extreme cases of this conjecture (d=0, m=n, m=1, m=0) have been proven.