No.

Monomial

Defining equation

1

1

${\phi }_t=0$

2

$u$

${\phi }_{xx}=0$

3

$u_x$

$-{\xi }_t-2{\phi }_x=0$

4

$u_x^2$

${\phi }_u-{\tau }_t-2{\phi }_u+2{\xi }_x=0$

5

$u_{xx}$

$-\phi =0$

6

$u{u}_x$

${\xi }_{xx}-2{\phi }_{xu}=0$

7

$u_x{u}_{xx}$

$-{\phi }_u+2{\xi }_x+{\phi }_u-{\tau }_t=0$

8

$u_{xx}^2$

$-{\tau }_u+{\tau }_u=0$

9

$u{u}_{xx}$

$-{\phi }_u+2{\xi }_x+{\phi }_u-{\tau }_t=0$

10

$u{u}_x{u}_{xx}$

$3{\xi }_u+2{\tau }_u=0$

11

$u{u}_{xt}$

$2{\tau }_x=0$

12

$u{u}_x{u}_{xt}$

$-2{\tau }_u=0$

13

$u_{xx}{u}_x^2$

$-{\tau }_{uu}=0$