In the authors’ articles [“Inverse scattering problem for vector fields and the heavenly equation”, (2005); arXiv:nlin/0512043 and Phys. Lett. A359 613–619 (2006; Zbl 1236.37042); arXiv:nlin/0604024], the authors have used their solution of the inverse spectral problem for 1-parameter families of vector fields for construction of formal solution of the Cauchy problem for a class of multidimensional integrable nonlinear PDEs. Here they apply this theory \(1^{\circ}\) to construct the nonlinear Riemann-Hilbert (RH) dressing for the 2-dimensional Toda equation \((exp(\varphi))_{tt}=\varphi_{\zeta_{11}\zeta_{12}}\), \(2^{\circ}\) to present the formal solution to the wave form of this equation \((exp(\varphi))_{tt}=\varphi_{xx}+\varphi_{yy}\), \(3^{\circ}\) to investigate the longtime behaviour of their solutions and \(4^{\circ}\) to characterize the class of spectral data required for the linearization of the RH problem associated with a class of implicit solutions of PDEs.