The paper considers the second order multi-dimensional differential delay equation
with a particular symmetric behaviour of the vector-function at and . A lower estimate for the number of periodic solutions of period in the system is given. The paper generalizes similar results derived for a like first order differential delay equation considered in [Z. M. Guo and J. S. Yu, J. Differ. Equations 218, No. 1, 15–35 (2005; Zbl 1095.34043)].