| $\xi =\tfrac{x}{{t}^{\tfrac{1}{2}}},\eta =y$ | |
| ${{\rm{\Omega }}}_{9}{V}_{4}+{V}_{5}$ | $u=\tfrac{f(\xi ,\eta )}{x}$ | ${\xi }^{3}{g}_{\xi }+4f+2\xi {g}_{\xi }f-2\xi {f}_{\xi }=0$ |
| $v={ln}(x)+g(\xi ,\eta )$ | ${\xi }^{3}{f}_{\xi \eta }-2\xi {f}_{\eta }{f}_{\xi }+4{f}_{\eta }f-2\xi {{ff}}_{\xi \eta }-4-2{\xi }^{3}{g}_{\xi \xi \xi }=0$ |
| $\xi =x,\eta =y-t$ | |
| ${{\rm{\Omega }}}_{12}{V}_{2}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $-{g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0$ |
| $v=g(\xi ,\eta )$ | $-{f}_{\eta \eta }+{f}_{\eta }{f}_{\xi }+{f}_{\xi \eta }f+{g}_{\xi \xi \xi }=0$ |
| $\xi =x,\eta =y$ | |
| ${{\rm{\Omega }}}_{13}{V}_{4}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $1-{g}_{\xi }f+{f}_{\xi }=0$ |
| $v=t+g(\xi ,\eta )$ | ${f}_{\eta }{f}_{\xi }+{f}_{\xi \eta }f+{g}_{\xi \xi \xi }=0$ |
| $\xi =x-y,\eta =t$ | |
| ${{\rm{\Omega }}}_{15}{V}_{1}+{V}_{2}$ | $u=f(\xi ,\eta )$ | ${g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0$ |
| $v=g(\xi ,\eta )$ | $-{f}_{\xi \eta }-{f}_{\xi }^{2}-{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$ |
| $\xi =y,\eta =x-t$ | |
| ${{\rm{\Omega }}}_{17}{V}_{1}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $-{g}_{\eta }-{g}_{\eta }f+{f}_{\eta }=0$ |
| $v=g(\xi ,\eta )$ | $-{f}_{\xi \eta }+{f}_{\xi }{f}_{\eta }+{{ff}}_{\xi \eta }+{g}_{\eta \eta \eta }=0$ |
| $\xi =\tfrac{x}{{y}^{\tfrac{1}{2}}},\eta =\tfrac{t}{{y}^{\tfrac{1}{2}}}$ | |
| ${{\rm{\Omega }}}_{18}{V}_{3}+{V}_{4}+{V}_{5}$ | $u=f(\xi ,\eta )$ | $\tfrac{1}{2}+\eta {g}_{\eta }-\eta {g}_{\xi }f+\eta {f}_{\xi }=0$ |
| $v=\tfrac{1}{2}{ln}(t)+g(\xi ,\eta )$ | $\xi {f}_{\xi \eta }+\eta {f}_{\eta \eta }+{f}_{\eta }+\xi {f}_{\xi }^{2}+\eta {f}_{\xi }{f}_{\eta }+\xi {{ff}}_{\xi \xi }+\eta {{ff}}_{\xi \eta }+{{ff}}_{\xi }-2{g}_{\eta \eta \eta }=0$ |
| $\xi =x,\eta =y-t$ | |
| ${{\rm{\Omega }}}_{21}{V}_{2}+{V}_{4}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $1-{g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0$ |
| $v=t+g(\xi ,\eta )$ | $-{f}_{\eta \eta }+{f}_{\xi }{f}_{\eta }+{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$ |
| $\xi =x-y,\eta =t$ | |
| ${{\rm{\Omega }}}_{22}{V}_{1}+{V}_{2}+{V}_{4}$ | $u=f(\xi ,\eta )$ | ${g}_{\eta }-f-{g}_{\xi }f+{f}_{\xi }=0$ |
| $v=x+g(\xi ,\eta )$ | $-{f}_{\xi \eta }-{f}_{\xi }^{2}-{{ff}}_{\xi \xi }+{g}_{\xi \xi \xi }=0$ |
| $\xi =x-y,\eta =y-t$ | |
| ${{\rm{\Omega }}}_{23}{V}_{1}+{V}_{2}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $-{g}_{\eta }-{{fg}}_{\xi }+{f}_{\xi }=0$ |
| $v=g(\xi ,\eta )$ | ${f}_{\xi \eta }-{f}_{\eta \eta }-{f}_{\xi }^{2}+{f}_{\xi }{f}_{\eta }-{{ff}}_{\xi \xi }+{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$ |
| $\xi =y,\eta =x-t$ | |
| ${{\rm{\Omega }}}_{24}{V}_{1}+{V}_{4}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $-{g}_{\eta }-{{fg}}_{\eta }+{f}_{\eta }=0$ |
| $v=x+g(\xi ,\eta )$ | $-{f}_{\xi \eta }+{f}_{\xi }{f}_{\eta }+{{ff}}_{\xi \eta }+{g}_{\eta \eta \eta }=0$ |
| $\xi =x-y,\eta =y-t$ | |
| ${{\rm{\Omega }}}_{25}{V}_{1}+{V}_{2}+{V}_{4}+{V}_{6}$ | $u=f(\xi ,\eta )$ | $-{g}_{\eta }-f-{{fg}}_{\xi }+{f}_{\xi }=0$ |
| $v=x+g(\xi ,\eta )$ | ${f}_{\xi \eta }-{f}_{\eta \eta }-{f}_{\xi }^{2}+{f}_{\xi }{f}_{\eta }-{{ff}}_{\xi \xi }+{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$ |