Let $G$ acts on $M$ and $(t,x,u)\in M$, $g\in G$. $$g.(t,x,u)=(T,X,U) $$ Let $$T=t+\lambda_1\\ X=\lambda_3 t+ x +\lambda_1 \lambda_3 + \lambda_2\\ U=u+\lambda_3$$ where $u=u(x,t)$ and $\lambda_i$ are constants.
Then the total differential of $U$ respect to $T$ and $X$ is $$U_T=D_T U= u_t-\lambda_3 u_x\\ U_X=D_X U=u_x$$
How these total derivatives is taken?
In the above question, the dual implicit differential operators are defined as follow $$ D_{X^i}=\sum_{j=1}^p W^i_jD_{x^j},\quad W^j_i=(D_{x^j}X^i)^{-1}. $$ where $D_{x^i}$ is the usual total derivative.