For the critical generalized KdV equation partial _t u + partial _x (partial _x^2 u + u^5)=0 on {mathbb {R}}, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q''+Q^5=Q. For any nu in (0,frac{1}{3}), there exist global (for tge 0) solutions of the equation with the asymptotic behavior u(t,x)=t-ν2Qt-ν(x-x(t))+w(t,x)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} u(t,x)= t^{-\\frac{\\nu }{2}} Q\\left( t^{-\\nu } (x-x(t))\\right) +w(t,x) \\end{aligned}$$\\end{document}where, for some c>0, x(t)∼ct1-2νand‖w(t)‖H1(x>12x(t))→0ast→+∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x(t)\\sim c t^{1-2\\nu } \\quad \\text{ and }\\quad \\Vert w(t)\\Vert _{H^1(x>\\frac{1}{2} x(t))} \\rightarrow 0\\quad \\text{ as } t\\rightarrow +\\infty . \\end{aligned}$$\\end{document}Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.