  N=6 superconformal Chern-Simons theory with gauge group U(M)xU(N)} is dual to
N M2-branes and (M-N) fractional M2-branes, equivalently, discrete 3-form
holonomy at C4/Zk orbifold singularity. We show that, much like its regular
counterpart of M=N, the theory at planar limit have integrability structure in
the conformal dimension spectrum of single trace operators. We first revisit
the Yang-Baxter equation for a spin chain system associated with the single
trace operators. We show that the integrability by itself does not preclude
parity symmetry breaking. We construct two-parameter family of parity
non-invariant, alternating spin chain Hamiltonian involving three-site
interactions between 4 and 4* of SU(4). At weak `t Hooft coupling, we study the
Chern-Simons theory perturbatively and calculate anomalous dimension of single
trace operators up to two loops. The computation is essentially parallel to the
regular case M=N. We find that resulting spin chain Hamiltonian matches with
the Hamiltonian derived from Yang-Baxter equation, but to the one preserving
parity symmetry. We give several intuitive explanations why the parity symmetry
breaking is not detected in the Chern-Simons spin chain Hamiltonian at
perturbative level. We suggest that open spin chain, associated with open
string excitations on giant gravitons or dibaryons, can detect discrete flat
holonomy and hence parity symmetry breaking through boundary field.
