  Fourth - order analogue to the second Painlev\'{e} equation is studied. This
equation has its origin in the modified Korteveg - de Vries equation of the
fifth order when we look for its self - similar solution. All power and non -
power expansions of the solutions for the fouth - order analogue to the second
Painlev\'{e} equation near points $z=0$ and $z=\infty$ are found by means of
the power geometry method. The exponential additions to solutions of the
equation studied are determined. Comparison of the expansions found with those
of the six Painlev\'{e} equations confirm the conjecture that the fourth -
order analogue to the second Painlev\'{e} equation defines new transcendental
functions.
