  We study critical properties of a Majorana chain in the presence of two
competing interactions of the shortest possible range. The obtained phase
diagram is very rich and contains nine different phases, including three
floating, two Ising, and four gapped phases. In addition we report a wide
variety of quantum phase transitions: the supersymmetric tri-critical Ising
lines; the Lifshitz critical line characterized by the dynamical critical
exponent $z=3$; Kosterlitz-Thouless transitions, and an exotic first order
transition between the floating and the gapped phases. However, the most
surprising feature of the obtained phase diagram is the emergence of the
commensurate line at which the floating phases collapses into direct
transition. We provide numerical evidences that the resulting multi-critical
point belongs to the universality class of the eight-vertex model. Implications
in the context of supersymmetric properties of the Majorana chain is briefly
discussed.
