  We study the existence and stability of Dirac nodal lines in
three-dimensional layered systems, whose layers individually have Dirac nodal
points protected by chiral (sublattice) symmetry. The model system we consider
is the rhombohedral stack of graphene layers with each layer subjected to a
uniform external potential that respects either AIII or BDI classes. From the
Hamiltonians in either classes, a pair of nontrivial spiraling Dirac nodal
lines can be derived. The results are reasonable in accord to the topological
classification of gapless phases for codimension $2$. The nodal lines approach
each other as the magnitude of the potential increases, revealing their
annihilation due to the fact that regarding the full system their topological
invariants are cancelled out.
