  We consider the ABF background underlying the eta-deformed AdS5 x S5 sigma
model. This background fails to satisfy the standard IIB supergravity equations
which indicates that the corresponding sigma model is not Weyl invariant, i.e.
does not define a critical string theory in the usual sense. We argue that the
ABF background should still correspond to UV finite theory on a flat 2d
world-sheet, implying that the eta-deformed model is scale invariant. This
property follows from the formal relation via T-duality between the
eta-deformed model and the one defined by an exact type IIB supergravity
solution that has 6 isometries albeit broken by a linear dilaton. We find that
the ABF background satisfies candidate type IIB scale invariance conditions
which for the R-R field strengths are of the second order in derivatives.
Surprisingly, we also find that the this background obeys an interesting
modification of the standard IIB supergravity equations that are first order in
derivatives of R-R fields. These modified equations explicitly depend on
Killing vectors of the ABF background and, although not universal, imply the
universal scale invariance conditions. Moreover, we show that it is precisely
the non-isometric dilaton of the T-dual solution that leads, after the
T-duality, to a modification of type II equations from their standard form. We
conjecture that the modified equations should follow from kappa-symmetry of the
eta-deformed model. All our observations apply also to eta-deformations of AdS3
x S3 and AdS2 x S2 models.
