  We investigate the one-loop spectral problem of $\gamma$-twisted, planar
$\mathcal{N}$=4 Super Yang-Mills theory in the double-scaling limit of
infinite, imaginary twist angle and vanishing Yang-Mills coupling constant.
This non-unitary model has recently been argued to be a simpler version of
full-fledged planar $\mathcal{N}$=4 SYM, while preserving the latter model's
conformality and integrability. We are able to derive for a number of sectors
one-loop Bethe equations that allow finding anomalous dimensions for various
subsets of diagonalizable operators. However, the non-unitarity of these
deformed models results in a large number of non-diagonalizable operators,
whose mixing is described by a very complicated structure of non-diagonalizable
Jordan blocks of arbitrarily large size and with a priori unknown generalized
eigenvalues. The description of these blocks by methods of integrability
remains unknown.
