On some developments of the $\overline\partial$-dressing method

Some developments of the $\overline\partial$-dressing method concerning an algebraic scheme of constructing integrable equations and construction of solutions with special properties are considered. It is demonstrated how the matrix $\mathbf{KP}$ equation appears from the scalar dressing and, more generally, how to construct the integrable system corresponding to an arbitrary triad of polynomials. Using the nonlocal $\overline\partial$-problem approach in $(2+1)$ dimensions, it is shown that the $\overline\partial$-problem with a shift and (for decreasing solutions) the Riemann problem with a shift naturally arise in $(1+1)$ dimensions. The Boussinesq equation and the first order relativistically-invariant systems are investigated. The developed approach allows one also to investigate the structure of the continuous spectrum and the inverse scattering problem for an arbitrary order ordinary differential operator on the infinite line.