  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
$\mathcal{M}_2$.
