  In this work we generalize the classical notion of a (compact) twistor line
in the period domain of compact complex tori. We introduce two new types of
lines, which are non-compact analytic curves in the period domain of complex
tori. We study the analytic properties of the compactifications of the curves,
the preservation of cohomology classes of type (1,1) along the curves and the
twistor path connectivity of the period domain by the curves of one of the new
types.
