On invariant surfaces in gene network models

We construct an invariant two-dimensional surface in the phase portrait of a certain six-dimensional dynamical system which is considered as a model for the circular gene network functioning. This invariant surface contains an equilibrium point $S_0$ of the system, and if $S_0$ is hyperbolic then this surface contains a cycle of the system. The conditions for the existence of a cycle of this and similar systems were obtained earlier.