  Let $\alpha$ be a Morse closed $1$-form of a smooth $n$-dimensional manifold
$M$. The zeroes of $\alpha$ of index $0$ or $n$ are called \emph{centers}. It
is known that every non-vanishing de Rham cohomology class $u$ contains a Morse
representative without centers. The result of this paper is the one-parameter
analogue of the last statement: every generic path $ (\alpha_t)_{t\in [0,1]}$
of closed $1$-forms in a fixed class $u\neq 0$ such that $\alpha_0, \alpha_1$
have no centers, can be modified relatively to its extremities to another such
path $ (\beta_t)_{t\in [0,1]}$ having no center at all.
