Authors: Paz Albares ; Pilar G. Estévez ; Alejandro González-Parra ; Paula del Olmo

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $\tau$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

• 22E70 - Applications of Lie groups to the sciences; explicit representations
• 35A24 - Methods of ordinary differential equations applied to PDEs
• 35Q53 - KdV equations (Korteweg-de Vries equations)
• 37K10 - Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
• 37K15 - Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
• 37K35 - Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems