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This article addresses the study of the complex version of the modifiedKorteweg-de Vries equation using two different approaches. Firstly, thesingular manifold method is applied in order to obtain the associated spectralproblem, binary Darboux transformations and $\tau$-functions. The second partconcerns the identification of the classical Lie symmetries for the spectralproblem. The similarity reductions associated to these symmetries allow us toderive the reduced spectral problems and first integrals for the ordinarydifferential equations arising from such reductions.

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