  We study the largest particle-number-preserving sector of the dilatation
operator in maximally supersymmetric gauge theory. After exploring one-loop
Bethe Ansatze for the underlying spin chain with psl(2|2) symmetry for simple
root systems related to several Kac-Dynkin diagrams, we use the analytic Bethe
Anzats to construct eigenvalues of transfer matrices with finite-dimensional
atypical representations in the auxiliary space. We derive closed Baxter
equations for eigenvalues of nested Baxter operators. We extend these
considerations for a non-distinguished root system with FBBF grading to all
orders of perturbation theory in 't Hooft coupling. We construct generating
functions for all transfer matrices with auxiliary space determined by Young
supertableaux (1^a) and (s) and find determinant formulas for transfer matrices
with auxiliary spaces corresponding to skew Young supertableaux. The latter
yields fusion relations for transfer matrices with auxiliary space
corresponding to representations labelled by square Young supertableaux. We
derive asymptotic Baxter equations which determine spectra of anomalous
dimensions of composite Wilson operators in noncompact psl(2|2) subsector of
N=4 super-Yang-Mills theory.
