This work concerns graph-operator models of spectrally embedded bound states or wave-guided modes that have been observed experimentally and numerically in optical systems. The reducibility of the Fermi surface of a periodic graph operator is necessary for a local defect to generate a spectrally embedded bound state that is spatially extended. The simplest construction of such states employs finite symmetries of the graph to produce invariant subspaces of the operator. This results in reducibility of the Fermi surface and symmetry-induced spectrally embedded bound states. But it has been observed that optical systems with no apparent symmetries can also admit spectrally embedded modes. I will show how graph operators can be constructed, in which the Fermi surface is separable and embedded eigenvalues are supported by local defects, but yet the eigenfunctions are not induced by any symmetry of the operator.