  We study cones and cylinders with a 1-parametric isometric deformation
carrying at least two planar curves, which remain planar during this continuous
flexion and are located in non-parallel planes. We investigate this
geometric/kinematic problem in the smooth and the discrete setting, as it is
the base for a generalized construction of so-called T-hedral zipper tubes. In
contrast to the cylindrical case, which can be solved easily, the conical one
is more tricky, but we succeed to give a closed form solution for the discrete
case, which is used to prove that these cones correspond to caps of Bricard
octahedra of the plane-symmetric type. For the smooth case we are able to
reduce the problem by means of symbolic computation to an ordinary differential
equation, but its solution remains an open problem.
