  Using chiral decomposition, we are able to find analytically the zero modes
and the conditions for such modes to exist in the Kitaev ladder model and
superconducting nanowires with Dresselhaus spin-orbit coupling. As a result, we
are able to calculate the number of zero modes in these systems for arbitrary
given parameters in the semi-infinite limit. Moreover, we find that when
suitable resonance condition is satisfied exact zero modes exist even in finite
systems contrary to the common belief.
