  We study certain new properties of 2D surfaces associated with the
$\mathbb{C}P^{N-1}$ models and the wave functions of the corresponding linear
spectral problem. We show that $su(N)$-valued immersion functions expressed in
terms of rank-1 orthogonal projectors are linearly dependent, but they span an
$(N-1)$-dimensional subspace of the Lie algebra $su(N)$. Their minimal
polynomials are cubic, except for the holomorphic and antiholomorphic
solutions, for which they reduce to quadratic trinomials. We also derive the
counterparts of these relations for the wave functions of the linear spectral
problems. In particular, we provide a relation between the wave functions,
which results from the partition of unity into the projectors. Finally, we show
that the angle between any two position vectors of the immersion functions,
corresponding to the same values of the independent variables, does not depend
on those variables.
