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Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that Mobius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary Backlund transformation. A more general class of multicomponent Mobius-type symmetries is studied. It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent Mobius-type symmetries give rise to Calogero–Moser system.

The initial value problem for the Harry Dym equation on the complex plane is linearised by the ∂ -method. A wide class of exact implicit solutions is constructed by the ∂ -dressing method. It is shown that the peculiarity of the HDC equation is connected with the presence of an essential singularity for the corresponding eigenfunction. The method to overcome this singularity is proposed. However, the implicitness of the exact solutions arises as a penalty.

The localized solutions for the Ishimori equation are studied. Using the general formulae derived in paper I and the exact solutions of the inverse problem for the modified Kadomtsev-Petviashvili equation we construct the exponentially and rationally localized soliton type solutions of the Ishimori equation with time dependent boundaries. Explicit examples of such solutions are presented.

The general form of the nonlinear evolution equations integrable by the linear matrix spectral problem and their Hamiltonian structures are presented for potentials with nonzero constant asymptotic values. The reduction problem is discussed within the framework of the AKNS method. A method is proposed for the calculation of the Hamiltonian structures for the integrable equations under reductions. Matrix ZN and N reductions and some other reductions are considered as examples. General forms of the corresponding reduced equations and their Hamiltonian structures are found.

The general form of the integrable equations and their Backlund transformations connected with the general two-dimensional Gelfand-Dikij-Zakharov-Shabat spectral problem is found within the framework of the generalized AKNS method. The bilocal tensor product of the solutions of the spectral problem is used successively, which essentially simplifies the calculations of recursion operators. The transformation properties of the integrable equations and Backlund transformations under the gauge group are discussed.

The treatment of Backlund transformations (BTS) in multi-dimensional spaces via gauge transformations is presented. The general two-dimensional N*N matrix Zakharov-Shabat-AKNS spectral problem and the general two-dimensional Nth-order Zakharov-Shabat-Gelfand-Dikij (ZSGD) spectral problem are considered in detail. The structure of the infinite-dimensional group of general BTS is studied. It is shown that one has N elementary BTS for the N*N matrix problem and only one elementary BT for the Nth-order ZSGD problem. The explicit formulae for soliton and elementary BTS for both spectral problems are found. Non-linear superposition formulae for elementary BTS for both problems are obtained. An infinite lattice of solutions of the integrable evolution equations related to the N*N matrix problem is constructed.