  In this paper we prove Implicit Function Theorems (IFT) for algebraic
varieties defined by regular quadratic equations and, more generally, regular
NTQ systems over free groups. In the model theoretic language these results
state the existence of very simple Skolem functions for particular
$\forall\exists$-formulas over free groups. We construct these functions
effectively. In non-effective form IFT first appeared in \cite{Imp}.
  From algebraic geometry view-point IFT can be described as lifting solutions
of equations into generic points of algebraic varieties.
  Moreover, we show that the converse is also true, i.e., IFT holds only for
algebraic varieties defined by regular NTQ systems. This implies that if a
finitely generated group $H$ is $\forall\exists$-equivalent to a free
non-abelian group then $H$ is isomorphic to the coordinate group of a regular
NTQ system.
