  For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove
existence of various degenerate solutions depending on the complex parameter
$t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$
and $t=o(x^{2/3})$. Using this result, we identify the most degenerate
solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,...,6$, called {\em
tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large
$n$ asymptotics of the coefficients in a formal expansion of these solutions.
We supplement our findings by a numerical study of the tritronqu\'ee solutions.
