We examine smooth four-dimensional vector fields reversible under somesmooth involution \(L\) that has a smooth two-dimensional submanifold of fixedpoints. Our main interest here is in the orbit structure of such a systemnear two types of heteroclinic connections involving saddle-foci andheteroclinic orbits connecting them. In both cases we found families ofsymmetric periodic orbits, multi-round heteroclinic connections andcountable families of homoclinic orbits of saddle-foci. All this suggests that the orbitstructure near such connections is very complicated. A non-variational version of the stationary Swift – Hohenberg equation is considered, as an example, where such structure has been found numerically.
