We study the tritronqu\'{e}e solution u(x,t) of the PI2β 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
xβΒ±β, 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.Comment: 23 pages, 3 figure