  It has been shown that the 4-dimensional Einstein-Maxwell-dilaton theory
allows a Bogomol'nyi-type inequality for an arbitrary dilaton coupling constant
$\alpha $, and that the bound is saturated if and only if the (asymptotically
flat) spacetime admits a nontrivial spinor satisfying the gravitino and the
dilatino Killing spinor equations. The present paper revisits this issue and
argues that the dilatino equation fails to ensure the dilaton field equation
unless the solution is purely electric/magnetic, or the dilaton coupling
constant is given by $\alpha=0, \sqrt 3$, corresponding to the
Brans-Dicke-Maxwell theory and the Kaluza-Klein reduction of 5-dimensional
vacuum gravity, respectively. A systematic classification of the supersymmetric
solutions reveals that the solution can be rotating if and only if the solution
is dyonic or the coupling constant is given by $\alpha=0, \sqrt 3$. This
implies that the theory with $\alpha \ne 0, \sqrt 3$ cannot be embedded into
supergravity except for the static truncation. Physical properties of
supersymmetric solutions are explored from various points of view.
