  Non-minimal scalar-tensor (ST) theories may admit an Einstein frame
representation, where gravity is described by the Einstein-Hilbert action plus
the scalar sector. Some STs become just {\em minimal} Einstein-scalar (MES)
theories, notable examples are Brans-Dicke and $R\phi^2$. ST theories with {\em
derivative} coupling can also be reduced to an Einstein frame by disformal
transformations, but, as a rule, their scalar sector will contain higher
derivative terms. Here we draw attention to a new Palatini kinetically coupled
theory which can be reduced to pure MES by an invertible disformal
transformation. This theory can then be converted into the Jordan frame of
another ST, $R\phi^2$, which also admits an invertible transformation to MES.
Two theories, each of which is dual to MES, will be {\em sequentially dual} to
each other and can be considered as two different Jordan frames of the same
theory. Both chosen theories violate null energy condition. Transforming the
same singular MES solutions, we find the desingularization signs in both Jordan
frames, but these are more pronouned in the framework of kinetic theory,
leading, in the cosmological case, to Genesis-type behavior.
