Define a Lie point symmetry of the first order ordinary differential equation $\Delta[t, \mathbf{x}, \dot{\mathbf{x}}]=$ 0. Describe such a Lie point symmetry in terms of the vector field that generates it.

Consider the $2 n$-dimensional Hamiltonian system $(M, H)$ governed by the differential equation

$$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=J \frac{\partial H}{\partial \mathbf{x}}$$

Define the Poisson bracket $\{\cdot, \cdot\}$. For smooth functions $f, g: M \rightarrow \mathbf{R}$ show that the associated Hamiltonian vector fields $V_{f}, V_{g}$ satisfy

$$\left[V_{f}, V_{g}\right]=-V_{\{f, g\}} .$$

If $F: M \rightarrow \mathbf{R}$ is a first integral of $(M, H)$, show that the Hamiltonian vector field $V_{F}$ generates a Lie point symmetry of $(\star)$. Prove the converse is also true if $(\star)$ has a fixed point, i.e. a solution of the form $\mathbf{x}(t)=\mathbf{x}_{0}$.