  Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each
node has a fundamental polynomial of degree $n.$ Assume that\\ $\#\mathcal
X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at
most three linearly independent curves of degree less than or equal to $n-1$
that pass through all the nodes of $\mathcal X.$ We provide a characterization
of the case when there are exactly three such curves. Namely, we prove that
then the set $\mathcal X$ has a special construction: either all its nodes
belong to a curve of degree $n-2,$ or all its nodes but three belong to a
(maximal) curve of degree $n-3.$
  This result complements a result established recently by H. Kloyan, D.
Voskanyan, and H. H. Note that the proofs of the two results are completely
different.
