Summary: The paper is devoted to the investigation of a triple-zero singularity and the generalized associated eigenspace. A reduced form for parameterized delay differential systems is obtained by using center manifold reduction and normal form calculation. The results given in the paper by

*E. Freire, E. Gamero, A. J. Rodríguez-Luis* and

*A. Algaba* [Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, No. 12, 2799–2820 (2002;

Zbl 1043.37042)] are employed to analyze the bifurcation behavior of the parameterized delay differential system with triple-zero singularity in detail and an example is presented to illustrate the results.