  We reconstruct the neutrino mass matrix in the flavor basis, using all
available experimental data on neutrino oscillations. Majorana nature of
neutrinos, normal mass hierarchy (ordering) and validity of the LMA MSW
solution of the solar neutrino problem are assumed. We study dependences of the
mass matrix elements, m_{alpha beta}, on the CP violating Dirac, delta, and
Majorana, rho, sigma, phases, for different values of the mixing angle
theta_{13} and of the absolute mass scale, m_1. The contours of constant mass
in the rho-sigma plane have been constructed for all m_{alpha beta}. These
rho-sigma plots allow to systematically scan all possible structures of the
mass matrix. We identify regions of parameters in which the matrix has (i) a
structure with the dominant mu tau-block, (ii) various hierarchical structures,
(iii) flavor alignment, (iv) structures with special ordering or equalities of
elements, (v) the democratic form. In certain cases the matrix can be
parameterized by powers of a unique expansion (ordering) parameter lambda
approximately 0.2 - 0.3 (lambda_{ord} approximately 0.6 - 0.7). Perspectives to
further restrict the structure of mass matrix in future experiments, in
particular in the beta beta_{0 nu}-decay searches, are discussed.
