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To investigate the degree $d$ connectedness locus, Thurston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in three articles. In the first one we construct the lamination $C_sCL$ together with the induced factor space $\mathbb{S}/C_sCL$ of the unit circle $\mathbb{S}$. As will be verified in the third paper, $\mathbb{S}/C_sCL$ is a monotone model of the \emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\mathcal{M}^{comb}_2$ of the Mandelbrot set $\mathcal{M}_2$., Comment: 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2201.11434

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f(z) = z^2 +c$. In the spirit of Thurston's work, we consider the space of all \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in a series of three articles. In the present paper, the first in the series, we construct a lamination $C_sCL$ together with the induced factor space ${\mathbb{S}}/C_sCL$ of the unit circle ${\mathbb{S}}$. As will be verified in the third paper of the series, ${\mathbb{S}}/C_sCL$ is a monotone model of the \emph{cubic symmetric connected locus}, i.e. the space of all cubic symmetric polynomials with connected Julia sets., Comment: 37 pages, 4 figures