= Solution
A <Majorana zero mode> is a normalized self-adjoint fermion operator localized near a boundary and commuting with the Hamiltonian. With open boundaries,
$$
H_0=it\sum_{j=1}^{M-1}c_{2j}c_{2j+1}.
$$
Neither $c_1$ nor $c_{2M}$ appears. Each anticommutes with both factors in every displayed bilinear and consequently commutes with their product. Thus
$$
\boxed{[H_0,c_1]=[H_0,c_{2M}]=0},
$$
and $c_1,c_{2M}$ are Majorana zero modes localized exactly at the two end sites.
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