  The recursion relations of branching coefficients $k_{\xi}^{(\mu)}$ for a
module $L_{\frak{g}\downarrow \frak{h}}^{\mu}$ reduced to a Cartan subalgebra
$\frak{h}$ are transformed in order to place the recursion shifts $\gamma \in
\Gamma_{\frak{a}\subset \frak{h}}$ into the fundamental Weyl chamber. The new
ensembles $F\Psi$ (the "folded fans") of shifts were constructed and the
corresponding recursion properties for the weights belonging to the fundamental
Weyl chamber were formulated. Being considered simultaneously for the set of
string functions (corresponding to the same congruence class $\Xi_{v}$ of
modules) the system of recursion relations constitute an equation
$\mathbf{M}_{(u)}^{\Xi_{v}} \mathbf{m}_{(u)}^{\mu}= \delta_{(u)}^{\mu}$ where
the operator $\mathbf{M}_{(u)}^{\Xi_{v}}$ is an invertible matrix whose
elements are defined by the coordinates and multiplicities of the shift weights
in the folded fans $F\Psi$ and the components of the vector
$\mathbf{m}_{(u)}^{\mu}$ are the string function coefficients for $L^{\mu}$
enlisted up to an arbitrary fixed grade $u$. The examples are presented where
the string functions for modules of $\frak{g}=A_{2}^{(1)}$ are explicitly
constructed demonstrating that the set of folded fans provides a compact and
effective tool to study the integrable highest weight modules.
