Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

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.