  We introduce a new 2-parameter family of sigma models exhibiting Poisson-Lie
T-duality on a quasitriangular Poisson-Lie group $G$. The models contain
previously known models as well as a new 1-parameter line of models having the
novel feature that the Lagrangian takes the simple form
$L=E(u^{-1}u_+,u^{-1}u_-)$ where the generalised metric $E$ is constant (not
dependent on the field $u$ as in previous models). We characterise these models
in terms of a global conserved $G$-invariance. The models on $G=SU_2$ and its
dual $G^\star$ are computed explicitly. The general theory of Poisson-Lie
T-duality is also extended; we develop the Hamiltonian formulation and the
reduction for constant loops to integrable motion on the group manifold.
Finally, we generalise T-duality in the Hamiltonian formulation to group
factorisations $D=G\dcross M$ where the subgroups need not be dual or even have
the same dimension and need not be connected to the Drinfeld double or to
Poisson structures.
