  For the mass 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 $\nu\in (0,\frac 13)$, there exist global (for
$t\geq 0$) solutions of the equation with the asymptotic behavior
\begin{equation*} u(t,x)= t^{-\frac{\nu}2} Q\left(t^{-\nu}
(x-x(t))\right)+w(t,x) \end{equation*} where, for some $c>0$, \begin{equation*}
x(t)\sim c t^{1-2\nu} \quad \mbox{and}\quad \|w(t)\|_{H^1(x>\frac 12 x(t))} \to
0\quad \mbox{as $t\to +\infty$.} \end{equation*} 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.
