  For the problems indicated in the title, a further development of a new
approach (different from those applied before) is given. A basic problem under
consideration arises in viscous incompressible fluid dynamics and describes
self-similar solutions to the boundary layer equation for a stream function
with zero pressure gradient (connected with the plane-parallel laminar flow in
a mixing layer). Some previous results concerning singular nonlinear Cauchy
problems, smooth stable initial manifolds, and parametric exponential Lyapunov
series are used to state correctly and analyze the singular
"initial-boundary-value" problem for a third-order nonlinear ordinary
differential equation defined on the entire real axis. The detailed analysis of
this singular nonlinear problem leads, in particular, to efficient methods for
solving it approximately and gives a possibility to obtain numerically the
particle trajectories in the plane of flow. Some results of the numerical
experiments are displayed and their physical interpretation is discussed. A
connection of this basic problem with some known physical and mathematical
problems, arising for self-similar solutions to the boundary layer equations
with zero pressure gradient, is described, namely the "flooded jet", the plane
"semi-jet" and the "near-wall jet" problems are considered which are of
interest by themselves.
