= Solution
For a holonomic conservative system the <Lagrangian> is $L(q,\dot q,t)=T-V$, the difference between <kinetic energy> and <potential energy>. The <Euler-Lagrange equations> are
$$
\frac d{dt}\frac{\partial L}{\partial\dot q_j}-\frac{\partial L}{\partial q_j}=0,\qquad j=1,\ldots,n.
$$
If $\partial L/\partial q_j=0$, this gives $d p_j/dt=0$ for the <generalized momentum> $p_j=\partial L/\partial\dot q_j$. Such a coordinate is a <cyclic coordinate>.
In <cylindrical coordinates> the squared speed is $\dot r^2+r^2\dot\theta^2+\dot z^2$. The holonomic constraint $z=r^2/a$ gives $\dot z=2r\dot r/a$, while gravity has <potential energy> $V=mgz$. Therefore
$$
\boxed{L=\frac m2\left[\left(1+\frac{4r^2}{a^2}\right)\dot r^2+r^2\dot\theta^2\right]-\frac{mg r^2}{a}}.
$$
There is no explicit $\theta$ dependence, so $\theta$ is a <cyclic coordinate> and
$$
\boxed{p_\theta=mr^2\dot\theta=\text{constant}}.
$$
This is <angular momentum> about the vertical symmetry axis. At $r=0$ the polar angle is singular, but the Cartesian <angular momentum> $m(x\dot y-y\dot x)$ remains well defined and has the same conserved value.
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