  In this paper, we study the cyclic vectors of the shift operator $S_b$ acting
on de Branges-Rovnyak space $\mathcal H(b)$ associated to a non-extreme point
of the closed unit ball of $H^\infty$. We highlight an interesting link with a
completeness problem that we study using the Cauchy transform. This enables us
to obtain some nice consequences on cyclicity.
