  We determine the constraints imposed on the 10d target superspace geometry by
the requirement of classical kappa-symmetry of the Green-Schwarz superstring.
In the type I case we find that the background must satisfy a generalization of
type I supergravity equations. These equations depend on an arbitrary vector
X_a and imply the one-loop scale invariance of the GS sigma model. In the
special case when X_a is the gradient of a scalar \phi (dilaton) one recovers
the standard type I equations equivalent to the 2d Weyl invariance conditions
of the superstring sigma model. In the type II case we find a generalized
version of the 10d supergravity equations the bosonic part of which was
introduced in arXiv:1511.05795. These equations depend on two vectors \X_a and
K_a subject to 1st order differential relations (with the equations in the
NS-NS sector depending only on the combination X_a = \X_a + K_a). In the
special case of K_a=0 one finds that \X_a=\d_a \phi and thus obtains the
standard type II supergravity equations. New generalized solutions are found if
K_a is chosen to be a Killing vector (and thus they exist only if the metric
admits an isometry). Non-trivial solutions of the generalized equations
describe K-isometric backgrounds that can be mapped by T-duality to type II
supergravity solutions with dilaton containing a linear isometry-breaking term.
Examples of such backgrounds appeared recently in the context of integrable
\eta-deformations of AdS_n x S^n sigma models. The classical kappa-symmetry
thus does not, in general, imply the 2d Weyl invariance conditions for the GS
sigma model (equivalent to type II supergravity equations) but only weaker
scale invariance type conditions.
