This manuscript introduces a framework that focuses on the singularity of a zero-double-Hopf system with $1:1$ resonance in general retarded differential equations (RDDs). Initially. practical algorithms are proposed to identify the zero-double-Hopf singularity and the associated generalized eigenspace that corresponds to zero and two pairs of purely imaginary eigenvalues. Subsequently. https://halohealthcarers.shop/product-category/toe-separators/