  We consider Einstein-Maxwell-dilaton gravity with charged dust and
interaction of the form $P(\chi) F_{\mu\nu} F^{\mu\nu}$, where $P(\chi)$ is an
arbitrary function of the dilaton field $\chi$ that can be normal or phantom.
For any regular $P(\chi)$, static configurations are possible with arbitrary
functions $g_{00} = \exp(2\gamma(x^i))$ ($i=1,2,3$) and $\chi = \chi(\gamma)$,
without any assumption of spatial symmetry. The classical Majumdar-Papapetrou
system is restored by putting $\chi =$ const. Among possible solutions are
black-hole (BH) and quasi-black-hole (QBH) ones. Some general results on BH and
QBH properties are deduced and confirmed by examples. It is found, in
particular, that asymptotically flat BHs and QBHs can exist with positive
energy densities of matter and both scalar and electromagnetic fields.
