  We introduce the notion of a regular quadratic equation and a regular NTQ
system over a free group. We prove the results that can be described as
Implicit function theorems for algebraic varieties corresponding to regular
quadratic and NTQ systems. We will also show that the Implicit function theorem
is true only for these varieties. In algebraic geometry such results would be
described as lifting solutions of equations into generic points. From the model
theoretic view-point we claim the existence of simple Skolem functions for
particular $\forall\exists$-formulas over free groups. Proving these theorems
we describe in details a new version of the Makanin-Razborov process for
solving equations in free groups. We also prove a weak version of the Implicit
function theorem for NTQ systems which is one of the key results in the
solution of the Tarski's problems about the elementary theory of a free group.
