Consider the vectorial Sturm-Liouville problem:
\[
{
−
y
(
x
)
+
P
(
x
)
y
(
x
)
=
λ
I
d
y
(
x
)
A
y
(
0
)
+
I
d
y
′
(
0
)
=
0
B
y
(
1
)
+
I
d
y
′
(
1
)
=
0
\left \{\begin {array}{l} -{\mathbf y}(x)+P(x){\mathbf y}(x) = \lambda I_{d}{\mathbf y}(x) A{\mathbf y}(0)+I_{d}{\mathbf y}’(0)={\mathbf 0} B{\mathbf y}(1)+I_{d}{\mathbf y}’(1)={\mathbf 0} \end {array}\right .
\]
where
P
(
x
)
=
[
p
i
j
(
x
)
]
i
,
j
=
1
d
P(x)=[p_{ij}(x)]_{i,j=1}^{d}
is a continuous symmetric matrix-valued function defined on
[
0
,
1
]
[0,1]
, and
A
A
and
B
B
are
d
×
d
d\times d
real symmetric matrices. An eigenfunction
y
(
x
)
{\mathbf y}(x)
of the above problem is said to be of type (CZ) if any isolated zero of its component is a nodal point of
y
(
x
)
{\mathbf y}(x)
. We show that when
d
=
2
d=2
, there are infinitely many eigenfunctions of type (CZ) if and only if
(
P
(
x
)
,
A
,
B
)
(P(x), A, B)
are simultaneously diagonalizable. This indicates that
(
P
(
x
)
,
A
,
B
)
(P(x), A, B)
can be reconstructed when all except a finite number of eigenfunctions are of type (CZ). The results supplement a theorem proved by Shen-Shieh (the second author) for Dirichlet boundary conditions. The proof depends on an eigenvalue estimate, which seems to be of independent interest.