  First, we discuss the non-Gaussian type of self-similar solutions to the
Navier-Stokes equations. We revisit a class of self-similar solutions which was
studied in Canonne-Planchon (1996). In order to shed some light on it, we study
self-similar solutions to the 1D Burgers equation in detail, completing the
most general form of similarity profiles that it can possibly possess. In
particular, on top of the well-known source-type solution we identify a
kink-type solution. It is represented by one of the confluent hypergeometric
functions, viz. Kummer's function $M.$
  For the 2D Navier-Stokes equations, on top of the celebrated Burgers vortex
we derive yet another solution to the associated Fokker-Planck equation. This
can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type
solution above. Some asymptotic properties of this kind of solution have been
worked out. Implications for the 3D Navier-Stokes equations are suggested.
  Second, we address an application of self-similar solutions to explore more
general kind of solutions. In particular, based on the source-type self-similar
solution to the 3D Navier-Stokes equations, we consider what we could tell
about more general solutions.
