  It is discussed in detail the complete mathematical model of gravitational
lensing on a single cosmic string (CS) of general shape and position with
respect to the line of sight. CS are one-dimensional extended objects assuredly
predicted by modern cosmology. The presence of CS changes the global geometry
of the Universe, could clarify the properties of the early Universe, including
inflation models, and could serve as a unique proof of higher-dimensional
theories. Despite the fact that CS have not yet been reliable detected, there
are several strong independent indications of the existence of the CS, based of
CMB analysis and search of gravitational lens chains with special properties.
However, early considered models of straight CS presented only a small fraction
of the general CS-configurations to be observed. Now we propose model which
could significantly increase the possibilities of CS observational search. It
is considered more realistic models have necessarily include the inclinations
and bends of the CS. Besides, the recent analysis of observational data on the
search for gravitational-lens candidates, shows a large number of pairs that
could be explained by the complex geometry of the CS.
