  On a compact Riemann surface (X) with finite punctures (P_1, \ldots, P_k), we
define toric curves as multi-valued, totally unramified holomorphic maps to
(\mathbb{P}^n) with monodromy in a maximal torus of ({\rm PSU}(n+1)).
\textit{Toric solutions} for the ({\rm SU}(n+1)) system on
$X\setminus\{P_1,\ldots, P_k\}$ are recognized by their associated {\it toric}
curves in (\mathbb{P}^n). We introduce a character n-ensemble as an (n)-tuple
of meromorphic one-forms with simple poles and purely imaginary periods,
generating toric curves on (X) minus finitely many points. We establish on $X$
a correspondence between character $n$-ensembles and toric solutions to the
({\rm SU}(n+1)) system with finitely many cone singularities. Our approach not
only broadens seminal solutions for up to two cone singularities on the Riemann
sphere, as classified by Jost-Wang (Int. Math. Res. Not., (6):277-290, 2002)
and Lin-Wei-Ye (Invent. Math., 190(1):169-207, 2012), but also advances beyond
the limits of Lin-Yang-Zhong's existence theorems (J. Differential Geom.,
114(2):337-391, 2020) by introducing a new solution class.
