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We study the tritronqu\'{e}e solution $u(x,t)$ of the $\mathrm{P}_{\rmI}^{2}$ equation, the second member of the Painlev\'{e} I hierarchy. Thissolution is pole-free on the real line and has various applications inmathematical 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 onthis result, we successfully derive the total integrals of $u(x,t)$ and theassociated Hamiltonian.

Asymptotics and total integrals of the $\mathrm{P}_{\rm I}^{2}$ tritronqu\'{e}e solution and its Hamiltonian