  We consider the mutual distribution of two linearly independent solutions
y_1(x) and y_2(x) of the 1D Schroedinger equation with a random potential.
Since individual distributions of $y_1$ and $y_2$ are log-normal, it is
naturally to suggest that their mutual distribution is also log-normal. Such
hypothesis is confirmed in the deep of the allowed and forbidden bands, but
failed near the initial band edge. The mechanism of deviations from the
log-normal form is elucidated, and the first correction to it is calculated.
The latter allows to demonstrate broadening of the spectral lines in the
universal conductance fluctuations. A lot of new information is obtained on the
phase transition in the distribution P(\psi), where \psi is a combined phase
entering the evolution equations. According to the previous publications, this
transition is related with appearance of the imaginary part of \psi at a
certain energy E_0, and is not accompanied by singularities in the system
resistance. The real sense of this transition consists in the change of
configuration of four Lyapunov exponents, which determine the general solution:
there are two pairs of complex-conjugated exponents for E>E_0, while for E<E_0
all exponents become real. Realization of two different configurations is
confirmed for energies in the deep of the allowed and forbidden bands; it
proves the existence of the singular point E_0 at the formal level. The phase
transition can be observed in optical systems, tracing the sign of the field in
a wave, when the coordinate is changed.
