Consider a positive integer
n
n
and
γ
1
>
−
1
,
⋯
,
γ
n
>
−
1
\gamma _1>-1,\cdots ,\gamma _n>-1
. Let
D
=
{
z
∈
C
:
|
z
|
>
1
}
D=\{z\in \mathbb {C}:|z|>1\}
, and let
(
a
i
j
)
n
×
n
(a_{ij})_{n\times n}
denote the Cartan matrix of
s
u
(
n
+
1
)
\frak {su}(n+1)
. Utilizing the ordinary differential equation of
(
n
+
1
)
(n+1)
th order around a singular source of
S
U
(
n
+
1
)
{SU}(n+1)
Toda system, as discovered by Lin-Wei-Ye [Invent. Math. 190 (2012), pp. 169–207], we precisely characterize a solution
(
u
1
,
⋯
,
u
n
)
(u_1,\cdots , u_n)
to the
S
U
(
n
+
1
)
{SU}(n+1)
Toda system
{
∂
2
u
i
∂
z
∂
z
¯
+
∑
j
=
1
n
a
i
j
e
u
j
a
m
p
;
=
π
γ
i
δ
0
on
D
−
1
2
∫
D
∖
{
0
}
e
u
i
d
z
∧
d
z
¯
a
m
p
;
>
∞
for all
i
=
1
,
⋯
,
n
\begin{equation*} \begin {cases} \frac {\partial ^2 u_i}{\partial z\partial \bar z}+\sum _{j=1}^n a_{ij} e^{u_j}&=\pi \gamma _i\delta _0\text { on } D\\ \frac {\sqrt {-1}}{2}\,\int _{D\backslash \{0\}} e^{u_{i} }{d}z\wedge {d}\bar z &> \infty \end{cases} \quad \text {for all}\quad i=1,\cdots , n \end{equation*}
using
(
n
+
1
)
(n+1)
holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each
1
≤
i
≤
n
1\leq i\leq n
,
0
0
represents the cone singularity with angle
2
π
(
1
+
γ
i
)
2\pi (1+\gamma _i)
for the metric
e
u
i
|
d
z
|
2
e^{u_i}|{d}z|^2
on
D
∖
{
0
}
D\backslash \{0\}
, which can be locally characterized by
(
n
−
1
)
(n-1)
non-vanishing holomorphic functions at
0
0
.