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