Consider a vector field

$$V=\alpha x \frac{\partial}{\partial x}+\beta t \frac{\partial}{\partial t}+\gamma v \frac{\partial}{\partial v}$$

on $\mathbb{R}^{3}$, where $\alpha, \beta$ and $\gamma$ are constants. Find the one-parameter group of transformations generated by this vector field.

Find the values of the constants $(\alpha, \beta, \gamma)$ such that $V$ generates a Lie point symmetry of the modified $\mathrm{KdV}$ equation ( $\mathrm{KKdV}$ )

$$v_{t}-6 v^{2} v_{x}+v_{x x x}=0, \quad \text { where } \quad v=v(x, t)$$

Show that the function $u=u(x, t)$ given by $u=v^{2}+v_{x}$ satisfies the KdV equation and find a Lie point symmetry of $\mathrm{KdV}$ corresponding to the Lie point symmetry of $\mathrm{mKdV}$ which you have determined from $V$.