  By a nondegenerate $k$-parameterized family $K$ of periodic solutions we
understand the situation when the geometric multiplicity of the multiplier +1
of the linearized on $K$ system equals to $k.$ Bifurcation of asymptotically
stable periodic solutions from $K$ is well studied in the literature and
different conditions have been proposeddepending on whether the algebraic
multiplicity of +1 is $k$ or not (by Malkin, Loud, Melnikov, Yagasaki). In this
paper we assume that the later is unknown. Asymptotic stability can not be
understood in this case, but we demonstrate that the information about
uniqueness of periodic solutions is still available. Moreover, we show that
differentiability of the right hand sides is not necessary for the results of
this kind and our theorems are proven under a kind of Lipschitz continuity.
