Fermions in the Background of Dilatonic Sphalerons

We discuss the properties and interpretation of a discrete sequence of a static spherically symmetric solutions of the Yang-Mills-dilaton theory. This sequence is parametrized by the number $n$ of zeros of a component of the gauge field potential. It is demonstrated that solutions with odd $n$ posses all the properties of the sphaleron. It is shown that there are normalizable fermion zero modes in the background of these solutions. The question of instability is critically analysed.