Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) RmKdV mKdV), are connected through a transformation that combines both Miura transformations. Lax pairs straightforwardly obtained by means aforementioned have also identified classical Lie symmetries RmKdV. Non-trivial similarity reductions computed they yield non-autonomous ordinary differential equations (ODEs), whose as consequence reductions.