  We study the localization of zeros of Cauchy transforms of discrete measures
on the real line. This question is motivated by the theory of canonical systems
of differential equations. In particular, we prove that the spaces of Cauchy
transforms having the localization property are in one-to-one correspondence
with the canonical systems of special type, namely, those whose Hamiltonians
consist only of indivisible intervals accumulating on the left. Various aspects
of the localization phenomena are studied in details. Connections with the
density of polynomials and other topics in analysis are discussed.
