The authors consider the multi-dimensional differential delay equation \[ x^{\,\prime}(t)=-f(x(t-1))-f(x(t-2))-\dots-f(x(t-2n+1)),\quad x\in\mathbb R^N,\tag{\(*\)} \] under the following basic assumptions

(i)

\(f\in C(\mathbb R^N,\mathbb R^N)\) and \(f\) is odd, \(f(-x)=-f(x)\; \forall x\in\mathbb R^N\);

(ii)

there exists a continuously differentiable function \(F\) satisfying \(F(0)=0\) and \(\nabla F\equiv f\);

(iii)

\[ f(x)=A_0x+o(|x|)\;\text{as}\; |x|\to 0, \quad f(x)=A_{\infty}x+o(|x|)\text{ as } |x|\to\infty, \] where \(A_0\) and \(A_{\infty}\) are symmetric \(N\times N\) matrices.

Sufficient conditions for the existence of periodic solutions of period \(4n\), including multiple periodic solutions, satisfying the additional “symmetry” condition \(x(t-2n)\equiv-x(t)\) are given. The paper is built upon and continues the prior work of the authors done for the particular case of \(n=1\) (see [J. Differ. Equations 218, No. 1, 15–35 (2005; Zbl 1095.34043)]). The pioneering work by J. L. Kaplan and J. A. Yorke of 1974, including a conjecture therein, is another motivation for this work (see [J. Math. Anal. Appl. 48, 317–324 (1974; Zbl 0293.34102)]).