I have calculated the generators of the cylindrical $KdV$ equation $$u_t+(u/2t)+uu_x+u_{xxx}=0,$$ but I got three generators, $$X_1=\partial_x,\\ X_2=2t^{1/2}\partial_x+\left(1/2t^{1/2}\right)\partial_u, \\ X_3=x\partial_x+3t\partial_t-2u\partial_u~.$$
But books and papers show that it has a fourth generator $$X_4=2xt^{1/2}\partial_x+4t^{3/2}\partial_t+(x/t^{1/2}-4ut^{1/2})\partial_u.$$ Can anybody provide me with a calculation of this?