State and prove Noether's theorem in Lagrangian mechanics.

Consider a Lagrangian

$$\mathcal{L}=\frac{1}{2} \frac{\dot{x}^{2}+\dot{y}^{2}}{y^{2}}-V\left(\frac{x}{y}\right)$$

for a particle moving in the upper half-plane $\left\{(x, y) \in \mathbb{R}^{2}, y>0\right\}$ in a potential $V$ which only depends on $x / y$. Find two independent first integrals.