  We continue the study of analytic functions in the unit disk of finite order
with arbitrary set of singular points on the unit circle, introduced in
\cite{FG}. The main focus here is made upon the inverse problem: the existence
of a function from this class with a given singular set and zero set subject to
certain Blaschke-type condition. We also discuss the local analog of the main
result from \cite{FG} similar to the standard local Blaschke condition for
analytic and bounded functions in the unit disk.
