Abstract：Theory of local Hamiltonian structures associated with differential-geometric Poisson brackets of first order was introduced by B.A. Dubrovin and S.P. Novikov in 1983. Corresponding Hamiltonian hydrodynamic type systems possess local Lagrangian representations. If hydrodynamic type systems are integrable by the Tsarev Generalised Hodograph Method, then the theory of corresponding local Hamiltonian structure is associated with the theory of orthogonal curvilinear coordinate nets. In this talk we consider the "mirrored" case (with respect to orthogonal curvilinear coordinate nets), which we call "anti-flat". We show that some integrable hydrodynamic type systems can possess local Lagrangian representations and corresponding nonlocal Hamiltonian structures, which are associated with the "anti-flat" case.
