  We analyze the geometry of positive and completely positive, trace preserving
Pauli maps that are fully determined by up to two distinct parameters. This
includes five classes of symmetric and non-invertible Pauli channels. Using the
Hilbert-Schmidt metric in the space of the Choi-Jamio\lkowski states, we
compute the relative volumes of entanglement breaking, time-local generated,
and divisible channels. Finally, we find the shapes of the complete positivity
regions in relation to the tetrahedron of all Pauli channels.
