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We consider the ABF background underlying the eta-deformed AdS5 x S5 sigmamodel. This background fails to satisfy the standard IIB supergravity equationswhich 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 theABF background should still correspond to UV finite theory on a flat 2dworld-sheet, implying that the eta-deformed model is scale invariant. Thisproperty follows from the formal relation via T-duality between theeta-deformed model and the one defined by an exact type IIB supergravitysolution that has 6 isometries albeit broken by a linear dilaton. We find thatthe ABF background satisfies candidate type IIB scale invariance conditionswhich for the R-R field strengths are of the second order in derivatives.Surprisingly, we also find that the this background obeys an interestingmodification of the standard IIB supergravity equations that are first order inderivatives of R-R fields. These modified equations explicitly depend onKilling vectors of the ABF background and, although not universal, imply theuniversal scale invariance conditions. Moreover, we show that it is preciselythe non-isometric dilaton of the T-dual solution that leads, after theT-duality, to a modification of type II equations from their standard form. Weconjecture that the modified equations should follow from kappa-symmetry of theeta-deformed model. All our observations apply also to eta-deformations of AdS3x S3 and AdS2 x S2 models.

Scale invariance of the eta-deformed AdS5 x S5 superstring, T-duality and modified type II equations