x,\hs{2ex}b_y=x\bar{h}_y+w_y.$$ The conservation law reads, $$\partial_y b_x+\partial_x b_y=0,\hs{2ex}\partial_x\bar{h}_y+\partial_y\bar{h}_x=0$$ which allows us to write the current $\bar{h}_y$ as, $$\bar{h}_y=-\partial_x w_y-\partial_y w_x.$$ From the commutation relations, we learn that $w_x$ is invariant under the Galilean boost. From the discussion on the two-point functions, we find $$\partial_{y_1}{\left\langle}w_x(x_1,y_1) w_x(x_2,y_2){\right\rangle}=0,\hs{2ex}{\left\langle}\partial_y w_x \partial_y w_x{\right\rangle}=0.$$ From our assumptions that the spectrum of the dilation operator is discrete and non-negative, the following equation is valid as an operator equation $$\partial_y w_x=0.$$ We can shift the currents without changing the canonical commutation relations $$\bar{h}_y\rightarrow \bar{h}_y+\partial_x w_y,\hs{2ex}\bar{h}_x\rightarrow \bar{h}_x-\partial_y w_y.$$ The $\bar{h}_x$ component must be changed at the same time to keep the conservation law intact. The similar shifts also happen in the currents $b_\m$. Under the above shift, we can set $$\bar{h}_y=0, \label{hbary}$$ such that \_y|[h]{}\_x=0 which implies that $\bar{h}_x$ is a function of $x$ $$\bar{h}_x=\bar{h}_x(x).$$ This leads to the existence of an infinite set of conserved charges, $$M_{\epsilon}=\int \epsilon(x)\bar{h}_x(x) dx, \label{Mcharges}$$ where $\epsilon(x)$ is an arbitrary smooth function $x$. It is easy to see that $M_1$ with $\epsilon=1$ actually generates the translation along $y$ direction, while $M_x$ with $\epsilon(x)=x$ is the boost generator. This is consistent with the discussion in the warped CFT literatures[@Hofman:2011zj; @Detournay:2012pc]. We should emphasize here that this infinite set of conserved charges are common in the 2d local Galilean field theories. Next let us turn to the dilation current. Depending on the weight $c$, we will consider $c=0$ and $c\neq 0$ separately.

### Special case: $c=0$

In this case, we have $$d_x=d y\bar{h}_x+s_x,\hs{2ex} d_y=dy\bar{h}_y+s_y.$$ The equations above can be taken