  We consider near horizon fall-off conditions of stationary black holes in
Einstein-Maxwell-Dilaton theory and find conserved charge conjugate to symmetry
generator that preserves near horizon fall-off conditions. Subsequently, we
find supertranslation, superrotation and multiple-charge modes. We apply the
obtained results on a typical static dilaton black hole and on a charged
rotating black string, as examples. In this case, supertranslation
double-zero-mode charge $\mathcal{T}_{(0,0)}$ is not equal to black hole
entropy times Hawking temperature. This may be seen as a problem but it is not,
because, in Einstein-Maxwell-Dilaton theory, we have a U(1) gauge freedom and
we use an appropriate gauge fixing to fix that problem. We show that new
entropy formula $4 \pi \hat{J}^{+}_{0} \hat{J}^{-}_{0}$, proposed in \cite{17},
is valid for black strings as well as black holes.
