= Solution
Use the axisymmetric streamfunction convention
$$
w=\frac1r\frac{\partial\psi}{\partial r},
\qquad
u=-\frac1r\frac{\partial\psi}{\partial z}.
$$
For the <similarity solution> $\psi=\nu z g(\eta)$, $\eta=r/z$, regularity on the axis requires $g(0)=g'(0)=0$, while matching to still ambient fluid requires $g'(\eta)/\eta\to0$ as $\eta\to\infty$. The given equation has the first integral
$$
\eta g'-2g+\frac12g^2=C.
$$
The axis conditions give $C=0$, and separation yields
$$
\frac{dg}{g(4-g)}=\frac{d\eta}{2\eta},
\qquad
g(\eta)=\frac{4a\eta^2}{1+a\eta^2}
$$
for some $a>0$.
The axial velocity is
$$
w=\frac{\nu}{\eta z}g'(\eta)
=\frac{8\nu a}{z(1+a\eta^2)^2}.
$$
Normalizing by the conserved momentum flux gives
$$
M=\int_0^\infty rw^2\,dr
=64\nu^2a^2\int_0^\infty\frac{\eta\,d\eta}{(1+a\eta^2)^4}
=\frac{32}{3}\nu^2a,
$$
so $a=3M/(32\nu^2)$. The <similarity solution for a laminar round jet> is therefore
$$
\boxed{g(\eta)=\frac{12M\eta^2}{32\nu^2+3M\eta^2}.}
$$
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