  The chiral Abelian Higgs model contains an interesting class of solitons
found by Rubakov and Tavkhelidze. These objects carry non-zero fermion number
$N_F$ (or Chern-Simons number $N_{CS}$, what is the same because of the chiral
anomaly) and are stable for sufficiently large $N_F$. In this paper we study
the properties of these anomalous solitons. We find that their
energy-versus-fermion-number ratio is given by $E\sim N_{CS}^{3/4}$ or $E\sim
N_{CS}^{2/3}$ depending on the structure of the scalar potential. For the
former case we demonstrate that there is a lower bound on the soliton energy,
which reads $E\geq c N_{CS}^{3/4}$, where $c$ is some parameter expressed
through the masses and coupling constants of the theory. We construct the
anomalous solitons numerically accounting both for Higgs and gauge dynamics and
show that they are not spherically symmetric. The thin wall approximation valid
for macroscopic solutions with $N_{CS} \gg 1$ is discussed as well.
