  In the absence of NSNS three-form flux the bosonic string on a symmetric
space is described by a symmetric space coset sigma-model. Such models are
known to be classically integrable. We show that the integrability extends also
to cases with non-zero NSNS flux (respecting the isometries) provided that the
flux satisfies a condition of the form H_{abc}H^{cde}~R_{ab}^{de}. We then turn
our attention to the type II Green-Schwarz superstring on a symmetric space. We
prove that if the space preserves some supersymmetry there exists a truncation
of the full superspace to a supercoset space and derive the general form of the
superisometry algebra. In the case of vanishing NSNS flux the corresponding
supercoset sigma-model for the string is known to be integrable. We prove that
the integrability extends to the full string by augmenting the supercoset Lax
connection with terms involving the fermions which are not captured by the
supercoset model. The construction is carried out to quadratic order in these
fermions. This proves the integrability of strings on symmetric spaces
supported by RR flux which preserve any non-zero amount of supersymmetry.
Finally we also construct Lax connections for some supercoset models with
non-zero NSNS flux describing strings in AdS(2,3) x S(2,3) x S(2,3) x T(2,3,4)
backgrounds preserving eight supersymmetries.
