Open questions about holomorphic solutions of soliton equations

We discuss the current situation with the following problems concerning local holomorphic solutions of soliton equations:
(1) Estimate for the order of the tau-function and its corollaries.
(2) Direct verification of the identities appearing from the Drinfeld–Sokolov reduction.
(3) Solubility of formal ${\mathrm U}(n)$-symmetric Riemann problems.
(4) Global boundedness of solutions with convergent Baker–Akhiezer function in the case of ${\mathrm U}(n)$-symmetry.
(5) Holomorphic extension to a wedge for solutions arising from the rapidly decaying inverse scattering approach.