  We study the tritronqu\'{e}e solution $u(x,t)$ of the $\mathrm{P}_{\rm
I}^{2}$ equation, the second member of the Painlev\'{e} I hierarchy. This
solution is pole-free on the real line and has various applications in
mathematical physics. We obtain a full asymptotic expansion of $u(x,t)$ as
$x\to\pm \infty$, uniformly for the parameter $t$ in a large interval. Based on
this result, we successfully derive the total integrals of $u(x,t)$ and the
associated Hamiltonian.
