Symmetric band complexes of thin type and chaotic sections which are not quite chaotic

In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic surface in the three-space with a plane direction such that the surface has a central symmetry, and the plane sections of the chosen direction are chaotic and consist of infinitely many connected components. Moreover, typical connected components of the sections have an asymptotic direction, which is due to the fact that the corresponding foliation on the surface in the 3-torus is not uniquely ergodic.