  One of the fourth-order analog to the first Painlev\'{e} equation is studied.
All power expansions for solutions of this equation near points $z=0$ and
$z=\infty$ are found by means of the power geometry method. The exponential
additions to the expansion of solution near $z=\infty$ are computed. The
obtained results confirm the hypothesis that the fourth-order analog of the
first Painlev\'{e} equation determines new transcendental functions.
