  We present and study ten-dimensional effective actions for various
non-geometric fluxes of which exotic branes act as the magnetic sources. Each
theory can be regarded as a U-dual version of the $\beta$-supergravity, a
reformulation of the ten-dimensional supergravity which is suitable for
describing non-geometric backgrounds with Q-flux. In each theory, we find a
solution that corresponds to the background of an exotic brane and show that it
is single-valued up to a gauge transformation, although the same background
written in the standard background fields is not single-valued in a usual
sense. Further, we also find a solution which corresponds to the background of
an instanton that is the electric dual of the exotic brane and discuss its
properties.
