To find maximum surface area for given arc length on a surface of revolution using cylindrical coordinates, we have to optimize
$$∫2π r \sqrt{dr^2+dz^2}+c1∫\sqrt{dr^2+dz^2+(rdθ)^2}$$
with Lagrangian form on $z$ as independent variable along symmetry axis :
$$ r \sqrt{1+\left(\dfrac{dr}{dz}\right)^2} −c ∫\sqrt{1+\left(\dfrac{dr}{dz}\right)^2 +\left(\dfrac{r d\theta}{dz}\right)^2}$$
EDIT1:
Let $\dfrac{ r d \theta }{ds}= \sin \psi $ and we have two constants respectively $\theta $, z basis.
$$ k_1=r \sin \psi,\quad k_2=\cos \phi\; (r- a \cos \psi)$$
in which the first invariant is known correct as Clairaut's constant and the second one is wrong.
Appreciate help to correctly obtain first integral Euler-Lagrange equation for with two or more independent variables.