A vectorial inverse nodal problem

Consider the vectorial Sturm-Liouville problem:

where $P(x)=[p_{ij}(x)]_{i,j=1}^{d}$ is a continuous symmetric matrix-valued function defined on $[0,1]$, and $A$ and $B$ are $d\times d$ real symmetric matrices. An eigenfunction ${\mathbf y}(x)$ of the above problem is said to be of type (CZ) if any isolated zero of its component is a nodal point of ${\mathbf y}(x)$. We show that when $d=2$, there are infinitely many eigenfunctions of type (CZ) if and only if $(P(x), A, B)$ are simultaneously diagonalizable. This indicates that $(P(x), A, B)$ 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.