= Solution
Balancing the time derivative, cubic nonlinearity and third spatial derivative gives $x\sim t^{1/3}$ and $u\sim t^{-1/3}$. Choose $\zeta=x/(3t)^{1/3}$ and $u=(3t)^{-1/3}w(\zeta)$. Then
$$
u_t=(3t)^{-4/3}(-w-\zeta w'),\quad
6u^2u_x=(3t)^{-4/3}6w^2w',\quad
u_{xxx}=(3t)^{-4/3}w''',\quad
u/t=(3t)^{-4/3}3w.
$$
Thus the <similarity solution> is
$$
\boxed{w'''+6w^2w'-\zeta w'+2w=0.}
$$
The extra spherical term changes the coefficient of $w$. This third-order equation cannot be integrated directly into the standard second-order <Painlevé II equation> of the ordinary modified KdV reduction.
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