@@ -1,498 +1,498 @@
-REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
-
-
- %Appendix (Testfile).
-
- %This appendix is a test file. The symmetry groups for various
- %equations or systems of equations are determined. The variable
- %PCLASS has the default value 0 and may be changed by the user
- %before running it. The output may be compared with the results
- %which are given in the references.
-
- %The Burgers equations
-
- deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
-
-
-
-  cresys deq 1$
-
- simpsys()$
-
- result()$
-
-
-The differential equation
-
-DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
-
-The symmetry generators are
-
-GEN(1):=dx(1)
-
-GEN(2):=dx(2)
-
-GEN(3):=dx(2)*x(1) + du(1)
-
-                  2
-GEN(4):=dx(1)*x(1)  + dx(2)*x(2)*x(1) + du(1)*( - u(1)*x(1) + x(2))
-
-GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
-
-The non-vanishing commutators of the finite subgroup
-
-COMM(1,3):= dx(2)
-
-COMM(1,4):= 2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
-
-COMM(1,5):= 2*dx(1)
-
-COMM(2,4):= dx(2)*x(1) + du(1)
-
-COMM(2,5):= dx(2)
-
-COMM(3,5):=  - dx(2)*x(1) - du(1)
-
-                           2
-COMM(4,5):=  - 2*dx(1)*x(1)
-
-          - 2*dx(2)*x(2)*x(1)
-
-          + 2*du(1)*(u(1)*x(1) - x(2))
-
-
- %The Kadomtsev-Petviashvili equation
-
- deq 1:=3*u(1,3,3)+u(1,2,2,2,2)+6*u(1,2,2)*u 1
-
-       +6*u(1,2)**2+4*u(1,1,2)$
-
-
-
-  cresys deq 1$
-
- simpsys()$
-
- result()$
-
-
-The differential equation
-
-DEQ(1):=3*u(1,3,3)
-
-        +u(1,2,2,2,2)
-
-        +6*u(1,2,2)*u(1)
-
-                 2
-        +6*u(1,2)
-
-        +4*u(1,1,2)
-
-The symmetry generators are
-
-GEN(1):=3*dx(2)*c(12) + 2*du(1)*df(c(12),x(1))
-
-GEN(2):= 6*dx(2)*df(c(9),x(1))*x(3)
-
-       - 9*dx(3)*c(9)
-
-       + 4*du(1)*df(c(9),x(1),2)*x(3)
-
-GEN(3):= 27*dx(1)*xi(1)
-
-                                          2
-       3*dx(2)*( - 2*df(xi(1),x(1),2)*x(3)  + 3*df(xi(1),x(1))*x(2))
-
-       + 18*dx(3)*df(xi(1),x(1))*x(3)
-
-       2*du(1)*(
-
-                                        2
-                -2*df(xi(1),x(1),3)*x(3)
-
-                +3*df(xi(1),x(1),2)*x(2)
-
-                -9*df(xi(1),x(1))*u(1))
-
-The remaining dependencies
-
-xi(1) depends on x(1)
-
-c(12) depends on x(1)
-
-c(9) depends on x(1)
-
-
-
-
- %The modified Kadomtsev-Petviashvili equation
-
-  deq 1:=u(1,1,2)-u(1,2,2,2,2)-3*u(1,3,3)
-
-       +6*u(1,2)**2*u(1,2,2)+6*u(1,3)*u(1,2,2)$
-
-
-
-  cresys deq 1$
-
- simpsys()$
-
- result()$
-
-
-The differential equation
-
-DEQ(1):=
-
-        -3*u(1,3,3)
-
-        +6*u(1,3)*u(1,2,2)
-
-        -u(1,2,2,2,2)
-
-                          2
-        +6*u(1,2,2)*u(1,2)
-
-        +u(1,1,2)
-
-The symmetry generators are
-
-GEN(1):=du(1)*c(16)
-
-GEN(2):=6*dx(2)*c(14) + du(1)*df(c(14),x(1))*x(3)
-
-GEN(3):= 12*dx(2)*df(c(11),x(1))*x(3)
-
-       + 72*dx(3)*c(11)
-
-                                        2
-          + du(1)*(df(c(11),x(1),2)*x(3)  + 6*df(c(11),x(1))*x(2))
-
-GEN(4):= 324*dx(1)*xi(1)
-
-                                         2
-        + 18*dx(2)*(df(xi(1),x(1),2)*x(3)  + 6*df(xi(1),x(1))*x(2))
-
-       + 216*dx(3)*df(xi(1),x(1))*x(3)
-
-                                              2
-           + du(1)*x(3)*(df(xi(1),x(1),3)*x(3)  + 18*df(xi(1),x(1),2)*x(2))
-
-The remaining dependencies
-
-xi(1) depends on x(1)
-
-c(16) depends on x(1)
-
-c(14) depends on x(1)
-
-c(11) depends on x(1)
-
-
-
-
- %The real- and the imaginary part of the nonlinear Schroedinger
- %equation
-
- deq 1:= u(1,1)+u(2,2,2)+2*u 1**2*u 2+2*u 2**3$
-
-
-
- deq 2:=-u(2,1)+u(1,2,2)+2*u 1*u 2**2+2*u 1**3$
-
-
-
- %Because this is not a single equation the two assignments
-
-  sder 1:=u(2,2,2)$
-
-  sder 2:=u(1,2,2)$
-
-
-
- %are necessary.
-
-  cresys()$
-
- simpsys()$
-
- result()$
-
-
-The differential equations
-
-DEQ(1):=u(2,2,2)
-
-               3
-        +2*u(2)
-
-                    2
-        +2*u(2)*u(1)
-
-        +u(1,1)
-
-DEQ(2):=
-
-        -u(2,1)
-
-               2
-        +2*u(2) *u(1)
-
-        +u(1,2,2)
-
-               3
-        +2*u(1)
-
-The symmetry generators are
-
-GEN(1):=dx(1)
-
-GEN(2):=dx(2)
-
-GEN(3):=du(1)*u(2) + du(2)*u(1)
-
-GEN(4):=2*dx(2)*x(1) - du(1)*u(2)*x(2) - du(2)*u(1)*x(2)
-
-GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1) + du(2)*u(2)
-
-The non-vanishing commutators of the finite subgroup
-
-COMM(1,4):= 2*dx(2)
-
-COMM(1,5):= 2*dx(1)
-
-COMM(2,4):=  - du(1)*u(2) - du(2)*u(1)
-
-COMM(2,5):= dx(2)
-
-COMM(3,5):=  - 2*du(1)*u(2) + 2*du(2)*u(1)
-
-COMM(4,5):=  - 2*dx(2)*x(1) + 3*du(1)*u(2)*x(2) - du(2)*u(1)*x(2)
-
-
- %The symmetries of the system comprising the four equations
-
-  deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
-
-
-
-  deq 2:=u(2,1)+u(2,2,2)$
-
-
-
-  deq 3:=u 1*u 2-2*u(2,2)$
-
-
-
-  deq 4:=4*u(2,1)+u 2*(u 1**2+2*u(1,2))$
-
-
-
-  sder 1:=u(1,2,2)$
-
- sder 2:=u(2,2,2)$
-
- sder 3:=u(2,2)$
-
- sder 4:=u(2,1)$
-
-
-
- %is obtained by calling
-
-  cresys()$
-
- simpsys()$
-
-Determining system is not completely solved
-
-
-The remaining equations are
-
-
-GL(1):=df(c(5),x(2),2) + df(c(5),x(1))
-
-GL(2):=df(c(5),x(2),x(1)) + df(c(5),x(2),3)
-
-
-The remaining dependencies
-
-c(5) depends on x(1),x(2)
-
-
-Number of functions is 21
-
-  df(c 5,x 1):=-df(c 5,x 2,2)$
-
-
-
-  df(c 5,x 2,x 1):=-df(c 5,x 2,3)$
-
-
-
-  simpsys()$
-
-  result()$
-
-
-The differential equations
-
-DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
-
-DEQ(2):=u(2,2,2) + u(2,1)
-
-DEQ(3):= - 2*u(2,2) + u(2)*u(1)
-
-                                            2
-DEQ(4):=4*u(2,1) + 2*u(2)*u(1,2) + u(2)*u(1)
-
-The symmetry generators are
-
-GEN(1):=dx(1)
-
-GEN(2):=dx(2)
-
-GEN(3):=du(2)*u(2)
-
-GEN(4):=2*dx(2)*x(1) + 2*du(1) + du(2)*u(2)*x(2)
-
-                     2
-GEN(5):= 4*dx(1)*x(1)
-
-       + 4*dx(2)*x(2)*x(1)
-
-      4*du(1)*( - u(1)*x(1) + x(2))
-
-                           2
-         + du(2)*u(2)*(x(2)  - 2*x(1))
-
-GEN(6):=4*dx(1)*x(1) + 2*dx(2)*x(2) - 2*du(1)*u(1) - du(2)*u(2)
-
-GEN(7):=du(1)*(2*df(c(5),x(2)) - c(5)*u(1)) + du(2)*c(5)*u(2)
-
-The remaining dependencies
-
-c(5) depends on x(1),x(2)
-
-
-
-Constraints
-
-
-df(c(5),x(1)):= - df(c(5),x(2),2)
-
-
-
-df(c(5),x(2),x(1)):= - df(c(5),x(2),3)
-
-
-
-The non-vanishing commutators of the finite subgroup
-
-COMM(1,4):= 2*dx(2)
-
-COMM(1,5):= 8*dx(1)*x(1) + 4*dx(2)*x(2) - 4*du(1)*u(1) - 2*du(2)*u(2)
-
-COMM(1,6):= 4*dx(1)
-
-COMM(2,4):= du(2)*u(2)
-
-COMM(2,5):= 4*dx(2)*x(1) + 4*du(1) + 2*du(2)*u(2)*x(2)
-
-COMM(2,6):= 2*dx(2)
-
-COMM(4,6):=  - 4*dx(2)*x(1) - 4*du(1) - 2*du(2)*u(2)*x(2)
-
-                            2
-COMM(5,6):=  - 16*dx(1)*x(1)
-
-          - 16*dx(2)*x(2)*x(1)
-
-          + 16*du(1)*(u(1)*x(1) - x(2))
-
-                               2
-          4*du(2)*u(2)*( - x(2)  + 2*x(1))
-
-
-
- %The symmetries of the subsystem comprising equation 1 and 3 are
- %obtained by
-
-  cresys(deq 1,deq 3)$
-
- simpsys()$
-
- result()$
-
-
-The differential equations
-
-DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
-
-DEQ(3):= - 2*u(2,2) + u(2)*u(1)
-
-The symmetry generators are
-
-GEN(1):=dx(1)
-
-GEN(2):=dx(2)
-
-GEN(3):=du(2)
-
-GEN(4):=2*dx(2)*x(1) + 2*du(1) + du(2)*x(2)
-
-GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
-
-                     2
-GEN(6):= 4*dx(1)*x(1)
-
-       + 4*dx(2)*x(2)*x(1)
-
-      4*du(1)*( - u(1)*x(1) + x(2))
-
-                     2
-         + du(2)*x(2)
-
-GEN(7):=du(2)*c(11)
-
-The remaining dependencies
-
-c(11) depends on x(1)
-
-
-
-The non-vanishing commutators of the finite subgroup
-
-COMM(1,4):= 2*dx(2)
-
-COMM(1,5):= 2*dx(1)
-
-COMM(1,6):= 8*dx(1)*x(1) + 4*dx(2)*x(2) - 4*du(1)*u(1)
-
-COMM(2,4):= du(2)
-
-COMM(2,5):= dx(2)
-
-COMM(2,6):= 4*dx(2)*x(1) + 4*du(1) + 2*du(2)*x(2)
-
-COMM(4,5):=  - 2*dx(2)*x(1) - 2*du(1) - du(2)*x(2)
-
-                        2
-COMM(5,6):= 8*dx(1)*x(1)
-
-          + 8*dx(2)*x(2)*x(1)
-
-         8*du(1)*( - u(1)*x(1) + x(2))
-
-                        2
-          + 2*du(2)*x(2)
-
-
- %The result for all possible subsystems is discussed in detail in
- %''Symmetries and Involution Systems: Some Experiments in Computer
- %Algebra'', contribution to the Proceedings of the Oberwolfach
- %Meeting on Nonlinear Evolution Equations, Summer 1986, to appear.
-
-end;
-(TIME:  spde 19070 19790)
+REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
+
+
+ %Appendix (Testfile).
+
+ %This appendix is a test file. The symmetry groups for various
+ %equations or systems of equations are determined. The variable
+ %PCLASS has the default value 0 and may be changed by the user
+ %before running it. The output may be compared with the results
+ %which are given in the references.
+
+ %The Burgers equations
+
+ deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
+
+
+
+  cresys deq 1$
+
+ simpsys()$
+
+ result()$
+
+
+The differential equation
+
+DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
+
+The symmetry generators are
+
+GEN(1):=dx(1)
+
+GEN(2):=dx(2)
+
+GEN(3):=dx(2)*x(1) + du(1)
+
+                  2
+GEN(4):=dx(1)*x(1)  + dx(2)*x(2)*x(1) + du(1)*( - u(1)*x(1) + x(2))
+
+GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
+
+The non-vanishing commutators of the finite subgroup
+
+COMM(1,3):= dx(2)
+
+COMM(1,4):= 2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
+
+COMM(1,5):= 2*dx(1)
+
+COMM(2,4):= dx(2)*x(1) + du(1)
+
+COMM(2,5):= dx(2)
+
+COMM(3,5):=  - dx(2)*x(1) - du(1)
+
+                           2
+COMM(4,5):=  - 2*dx(1)*x(1)
+
+          - 2*dx(2)*x(2)*x(1)
+
+          + 2*du(1)*(u(1)*x(1) - x(2))
+
+
+ %The Kadomtsev-Petviashvili equation
+
+ deq 1:=3*u(1,3,3)+u(1,2,2,2,2)+6*u(1,2,2)*u 1
+
+       +6*u(1,2)**2+4*u(1,1,2)$
+
+
+
+  cresys deq 1$
+
+ simpsys()$
+
+ result()$
+
+
+The differential equation
+
+DEQ(1):=3*u(1,3,3)
+
+        +u(1,2,2,2,2)
+
+        +6*u(1,2,2)*u(1)
+
+                 2
+        +6*u(1,2)
+
+        +4*u(1,1,2)
+
+The symmetry generators are
+
+GEN(1):=3*dx(2)*c(12) + 2*du(1)*df(c(12),x(1))
+
+GEN(2):= 6*dx(2)*df(c(9),x(1))*x(3)
+
+       - 9*dx(3)*c(9)
+
+       + 4*du(1)*df(c(9),x(1),2)*x(3)
+
+GEN(3):= 27*dx(1)*xi(1)
+
+                                          2
+       3*dx(2)*( - 2*df(xi(1),x(1),2)*x(3)  + 3*df(xi(1),x(1))*x(2))
+
+       + 18*dx(3)*df(xi(1),x(1))*x(3)
+
+       2*du(1)*(
+
+                                        2
+                -2*df(xi(1),x(1),3)*x(3)
+
+                +3*df(xi(1),x(1),2)*x(2)
+
+                -9*df(xi(1),x(1))*u(1))
+
+The remaining dependencies
+
+xi(1) depends on x(1)
+
+c(12) depends on x(1)
+
+c(9) depends on x(1)
+
+
+
+
+ %The modified Kadomtsev-Petviashvili equation
+
+  deq 1:=u(1,1,2)-u(1,2,2,2,2)-3*u(1,3,3)
+
+       +6*u(1,2)**2*u(1,2,2)+6*u(1,3)*u(1,2,2)$
+
+
+
+  cresys deq 1$
+
+ simpsys()$
+
+ result()$
+
+
+The differential equation
+
+DEQ(1):=
+
+        -3*u(1,3,3)
+
+        +6*u(1,3)*u(1,2,2)
+
+        -u(1,2,2,2,2)
+
+                          2
+        +6*u(1,2,2)*u(1,2)
+
+        +u(1,1,2)
+
+The symmetry generators are
+
+GEN(1):=du(1)*c(16)
+
+GEN(2):=6*dx(2)*c(14) + du(1)*df(c(14),x(1))*x(3)
+
+GEN(3):= 12*dx(2)*df(c(11),x(1))*x(3)
+
+       + 72*dx(3)*c(11)
+
+                                        2
+          + du(1)*(df(c(11),x(1),2)*x(3)  + 6*df(c(11),x(1))*x(2))
+
+GEN(4):= 324*dx(1)*xi(1)
+
+                                         2
+        + 18*dx(2)*(df(xi(1),x(1),2)*x(3)  + 6*df(xi(1),x(1))*x(2))
+
+       + 216*dx(3)*df(xi(1),x(1))*x(3)
+
+                                              2
+           + du(1)*x(3)*(df(xi(1),x(1),3)*x(3)  + 18*df(xi(1),x(1),2)*x(2))
+
+The remaining dependencies
+
+xi(1) depends on x(1)
+
+c(16) depends on x(1)
+
+c(14) depends on x(1)
+
+c(11) depends on x(1)
+
+
+
+
+ %The real- and the imaginary part of the nonlinear Schroedinger
+ %equation
+
+ deq 1:= u(1,1)+u(2,2,2)+2*u 1**2*u 2+2*u 2**3$
+
+
+
+ deq 2:=-u(2,1)+u(1,2,2)+2*u 1*u 2**2+2*u 1**3$
+
+
+
+ %Because this is not a single equation the two assignments
+
+  sder 1:=u(2,2,2)$
+
+  sder 2:=u(1,2,2)$
+
+
+
+ %are necessary.
+
+  cresys()$
+
+ simpsys()$
+
+ result()$
+
+
+The differential equations
+
+DEQ(1):=u(2,2,2)
+
+               3
+        +2*u(2)
+
+                    2
+        +2*u(2)*u(1)
+
+        +u(1,1)
+
+DEQ(2):=
+
+        -u(2,1)
+
+               2
+        +2*u(2) *u(1)
+
+        +u(1,2,2)
+
+               3
+        +2*u(1)
+
+The symmetry generators are
+
+GEN(1):=dx(1)
+
+GEN(2):=dx(2)
+
+GEN(3):=du(1)*u(2) + du(2)*u(1)
+
+GEN(4):=2*dx(2)*x(1) - du(1)*u(2)*x(2) - du(2)*u(1)*x(2)
+
+GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1) + du(2)*u(2)
+
+The non-vanishing commutators of the finite subgroup
+
+COMM(1,4):= 2*dx(2)
+
+COMM(1,5):= 2*dx(1)
+
+COMM(2,4):=  - du(1)*u(2) - du(2)*u(1)
+
+COMM(2,5):= dx(2)
+
+COMM(3,5):=  - 2*du(1)*u(2) + 2*du(2)*u(1)
+
+COMM(4,5):=  - 2*dx(2)*x(1) + 3*du(1)*u(2)*x(2) - du(2)*u(1)*x(2)
+
+
+ %The symmetries of the system comprising the four equations
+
+  deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
+
+
+
+  deq 2:=u(2,1)+u(2,2,2)$
+
+
+
+  deq 3:=u 1*u 2-2*u(2,2)$
+
+
+
+  deq 4:=4*u(2,1)+u 2*(u 1**2+2*u(1,2))$
+
+
+
+  sder 1:=u(1,2,2)$
+
+ sder 2:=u(2,2,2)$
+
+ sder 3:=u(2,2)$
+
+ sder 4:=u(2,1)$
+
+
+
+ %is obtained by calling
+
+  cresys()$
+
+ simpsys()$
+
+Determining system is not completely solved
+
+
+The remaining equations are
+
+
+GL(1):=df(c(5),x(2),2) + df(c(5),x(1))
+
+GL(2):=df(c(5),x(2),x(1)) + df(c(5),x(2),3)
+
+
+The remaining dependencies
+
+c(5) depends on x(1),x(2)
+
+
+Number of functions is 21
+
+  df(c 5,x 1):=-df(c 5,x 2,2)$
+
+
+
+  df(c 5,x 2,x 1):=-df(c 5,x 2,3)$
+
+
+
+  simpsys()$
+
+  result()$
+
+
+The differential equations
+
+DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
+
+DEQ(2):=u(2,2,2) + u(2,1)
+
+DEQ(3):= - 2*u(2,2) + u(2)*u(1)
+
+                                            2
+DEQ(4):=4*u(2,1) + 2*u(2)*u(1,2) + u(2)*u(1)
+
+The symmetry generators are
+
+GEN(1):=dx(1)
+
+GEN(2):=dx(2)
+
+GEN(3):=du(2)*u(2)
+
+GEN(4):=2*dx(2)*x(1) + 2*du(1) + du(2)*u(2)*x(2)
+
+                     2
+GEN(5):= 4*dx(1)*x(1)
+
+       + 4*dx(2)*x(2)*x(1)
+
+      4*du(1)*( - u(1)*x(1) + x(2))
+
+                           2
+         + du(2)*u(2)*(x(2)  - 2*x(1))
+
+GEN(6):=4*dx(1)*x(1) + 2*dx(2)*x(2) - 2*du(1)*u(1) - du(2)*u(2)
+
+GEN(7):=du(1)*(2*df(c(5),x(2)) - c(5)*u(1)) + du(2)*c(5)*u(2)
+
+The remaining dependencies
+
+c(5) depends on x(1),x(2)
+
+
+
+Constraints
+
+
+df(c(5),x(1)):= - df(c(5),x(2),2)
+
+
+
+df(c(5),x(2),x(1)):= - df(c(5),x(2),3)
+
+
+
+The non-vanishing commutators of the finite subgroup
+
+COMM(1,4):= 2*dx(2)
+
+COMM(1,5):= 8*dx(1)*x(1) + 4*dx(2)*x(2) - 4*du(1)*u(1) - 2*du(2)*u(2)
+
+COMM(1,6):= 4*dx(1)
+
+COMM(2,4):= du(2)*u(2)
+
+COMM(2,5):= 4*dx(2)*x(1) + 4*du(1) + 2*du(2)*u(2)*x(2)
+
+COMM(2,6):= 2*dx(2)
+
+COMM(4,6):=  - 4*dx(2)*x(1) - 4*du(1) - 2*du(2)*u(2)*x(2)
+
+                            2
+COMM(5,6):=  - 16*dx(1)*x(1)
+
+          - 16*dx(2)*x(2)*x(1)
+
+          + 16*du(1)*(u(1)*x(1) - x(2))
+
+                               2
+          4*du(2)*u(2)*( - x(2)  + 2*x(1))
+
+
+
+ %The symmetries of the subsystem comprising equation 1 and 3 are
+ %obtained by
+
+  cresys(deq 1,deq 3)$
+
+ simpsys()$
+
+ result()$
+
+
+The differential equations
+
+DEQ(1):=u(1,2,2) + u(1,2)*u(1) + u(1,1)
+
+DEQ(3):= - 2*u(2,2) + u(2)*u(1)
+
+The symmetry generators are
+
+GEN(1):=dx(1)
+
+GEN(2):=dx(2)
+
+GEN(3):=du(2)
+
+GEN(4):=2*dx(2)*x(1) + 2*du(1) + du(2)*x(2)
+
+GEN(5):=2*dx(1)*x(1) + dx(2)*x(2) - du(1)*u(1)
+
+                     2
+GEN(6):= 4*dx(1)*x(1)
+
+       + 4*dx(2)*x(2)*x(1)
+
+      4*du(1)*( - u(1)*x(1) + x(2))
+
+                     2
+         + du(2)*x(2)
+
+GEN(7):=du(2)*c(11)
+
+The remaining dependencies
+
+c(11) depends on x(1)
+
+
+
+The non-vanishing commutators of the finite subgroup
+
+COMM(1,4):= 2*dx(2)
+
+COMM(1,5):= 2*dx(1)
+
+COMM(1,6):= 8*dx(1)*x(1) + 4*dx(2)*x(2) - 4*du(1)*u(1)
+
+COMM(2,4):= du(2)
+
+COMM(2,5):= dx(2)
+
+COMM(2,6):= 4*dx(2)*x(1) + 4*du(1) + 2*du(2)*x(2)
+
+COMM(4,5):=  - 2*dx(2)*x(1) - 2*du(1) - du(2)*x(2)
+
+                        2
+COMM(5,6):= 8*dx(1)*x(1)
+
+          + 8*dx(2)*x(2)*x(1)
+
+         8*du(1)*( - u(1)*x(1) + x(2))
+
+                        2
+          + 2*du(2)*x(2)
+
+
+ %The result for all possible subsystems is discussed in detail in
+ %''Symmetries and Involution Systems: Some Experiments in Computer
+ %Algebra'', contribution to the Proceedings of the Oberwolfach
+ %Meeting on Nonlinear Evolution Equations, Summer 1986, to appear.
+
+end;
+(TIME:  spde 19070 19790)