Can someone explain which Weierstrass representation Emmy Noether refers to below?
As context, Noether's second Conservation Theorem says if a Lagrangian has an infinite-dimensional Lie group of symmetries parameterized by $r$ arbitrary functions, then the corresponding Euler-Lagrange expressions and their derivatives up to order $s$ satisfy $r$ independent equations--and conversely equations on the Euler-Lagrange expressions imply symmetries of the Lagrangian. She says:
The simplest example for Theorem II -- without its converse -- is Weierstrass’s parametric representation. Here, as is well known, the integral is invariant in the case of homogeneity of the first order when one replaces the independent variable $x$ by an arbitrary function of $x$ which leaves $u$ unchanged ($y = p(x)$; $v_i(y) = u_i(x)$). Thus an arbitrary function occurs though none of its derivatives occurs, and to this corresponds the well-known linear relation among the Lagrangian expressions themselves $\sum \psi_i\frac{du_i}{dx}=0$.
(The translation is from Yvette Kosmann-Schwarzbach's book The Noether Theorems p. 6.)