  We calculate the self-energy anomaly of a pointlike electric dipole located
in a static $(2+1)$-dimensional curved spacetime. The energy functional for
this problem is invariant under an infinite-dimensional (gauge) group of
transformations parameterized by one scalar function of two variables. We
demonstrate that the problem of the calculation of the self-energy anomaly for
a pointlike dipole can be reduced to the calculation of quantum fluctuations of
an effective two-dimensional Euclidean quantum field theory. We reduced the
problem in question to the calculation of the conformal anomaly of an effective
scalar field in two dimensions and obtained an explicit expression for the
self-energy anomaly of an electric dipole in an asymptotically flat, regular
$(2+1)$-dimensional spacetime which may have electrically neutral
black-hole-like metrics with regular Killing horizon.
