= Solution
For the metric
$$
du^2+Y(u)^2\,dv^2,
$$
the Christoffel symbols needed in the $v$ equation are
$$
\Gamma^v_{uv}=\Gamma^v_{vu}=\frac{Y'}Y.
$$
Thus a geodesic obeys
$$
\ddot v+2\frac{Y'}Y\dot u\dot v=0.
$$
Multiplication by $Y^2$ turns this into
$$
\frac d{dt}\left(Y(u(t))^2\dot v(t)\right)=0.
$$
Consequently the <Clairaut first integral for a surface of revolution> is
$$
\boxed{Y(u)^2\dot v=\text{constant}}.
$$
It is also the conserved momentum associated with the cyclic coordinate $v$.
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