  In this work, we show that an in-plane magnetic field can drive a fully
gapped $p \pm i p$ topological superconductor into a gapless phase which
supports symmetry protected Majorana edge states (MESs). Specifically, an
in-plane magnetic field can close the bulk gap and create zero energy Majorana
flat bands (MFBs) in the excitation spectrum. We show that the MFBs in the
gapless regime are protected by symmetry and are associated with MESs. The MFBs
acquire finite slopes when s-wave pairing and Rashba spin-oribit coupling terms
are added to the Hamiltonian. In this case, novel uni-directional MESs which
propagate in the same direction on opposite edges may appear. Uni-directional
MESs can also be found in pure s-wave superconductors with spin-orbit coupling.
The MFBs and the uni-directional MESs induce nearly quantized zero bias
conductance in tunneling experiments even in the presence of a gapless bulk and
disorder.
