An account of the Poisson-Lie T-duality is presented for the case when
the action of the duality group on a target is not free. At the same
time a generalization of the picture is given when the duality group d
oes not even act on sigma-model targets but only on their phase spaces
. The outcome is a huge class of dualizable targets generically having
no local isometries or Poisson-Lie symmetries whatsoever.