Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlev\'{e} equation

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.