Solution 1 to problem e3c2new
Remaining equations |
Expressions |
Parameters |
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Equations
The following unsolved equations remain:
2 2
0=b31 + b32
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
- b31*n2
n1=-----------
b32
- b32*m2
m1=-----------
b31
10 2 4 2
----*b31 *n2 + ---*b32 *n2
3 3
m3=----------------------------
a33*b32
104 2 2 56 2 2
- 16*a33*b32*k1*m2*n2*n3 - -----*b31 *k1*n2*n3 - ----*b32 *k1*n2*n3
3 3
r6=------------------------------------------------------------------------
4
a33 *b32
2 104 2 2 56 2 2
- 16*a33*b32*k1*m2*n2 - -----*b31 *k1*n2 *n3 - ----*b32 *k1*n2 *n3
3 3
r5=----------------------------------------------------------------------
4
a33 *b32
2 104 3 2 56 2 2
16*a33*b31*b32*k1*m2*n2 + -----*b31 *k1*n2 *n3 + ----*b31*b32 *k1*n2 *n3
3 3
r4=---------------------------------------------------------------------------
4 2
a33 *b32
r3=0
r2=0
r1=0
4 8 2 2 112 4
40*b31 *k1*m2*n2 + ---*b31 *b32 *k1*m2*n2 - -----*b32 *k1*m2*n2
3 3
q20=-----------------------------------------------------------------
3 3
a33 *b32
4 8 2 2 112 4
- 40*b31 *k1*m2*n2 - ---*b31 *b32 *k1*m2*n2 + -----*b32 *k1*m2*n2
3 3
q19=--------------------------------------------------------------------
3 2
a33 *b31*b32
64 2 2 64 2 2
8*a33*b32*k1*m2*n3 - ----*b31 *k1*n2 - ----*b32 *k1*n2
3 3
q17=----------------------------------------------------------
3
a33 *b32
3 160 4 2 160 2 2 2
- 8*a33*b32 *k1*m2*n3 + -----*b31 *k1*n2 + -----*b31 *b32 *k1*n2
3 3
q16=---------------------------------------------------------------------
3 2
a33 *b31*b32
2 2 2 2 2 4 2 580 6 2
q15=(4*a33 *b31 *b32 *k1*m2 + 4*a33 *b32 *k1*m2 - -----*b31 *k1*n2
9
704 4 2 2 124 2 4 2 4 2 2
- -----*b31 *b32 *k1*n2 - -----*b31 *b32 *k1*n2 )/(a33 *b31 *b32 )
9 9
5 2 3 2 2 4 2
40*b31 *k1*n2 + 64*b31 *b32 *k1*n2 + 24*b31*b32 *k1*n2
q14=-----------------------------------------------------------
4 3
a33 *b32
2 2
16*b31 *k1*n2
q13=----------------
3
a33 *b32
16 2 8 2
16*a33*b32*k1*m2*n2 - ----*b31 *k1*n2*n3 + ---*b32 *k1*n2*n3
3 3
q12=--------------------------------------------------------------
3
a33 *b32
16*a33*b31*k1*m2*n2 - 8*b31*b32*k1*n2*n3
q11=------------------------------------------
3
a33 *b32
2 2 2 2 2 4 2 256 6 2
q10=(4*a33 *b31 *b32 *k1*m2 + 4*a33 *b32 *k1*m2 - -----*b31 *k1*n2
9
200 4 2 2 56 2 4 2 4 2 2
- -----*b31 *b32 *k1*n2 + ----*b31 *b32 *k1*n2 )/(a33 *b31 *b32 )
9 9
3 2 2 2
16*b31 *k1*n2 + 32*b31*b32 *k1*n2
q9=-------------------------------------
3 2
a33 *b32
2
16*a33*b32*k1*m2*n2 + 8*b31 *k1*n2*n3
q8=---------------------------------------
3
a33 *b31
56 2 80 2
16*a33*b32*k1*m2*n2 + ----*b31 *k1*n2*n3 + ----*b32 *k1*n2*n3
3 3
q7=---------------------------------------------------------------
3
a33 *b32
8*k1*n2*n3
q5=------------
2
a33
- 8*b31*k1*n2*n3
q4=-------------------
2
a33 *b32
2 2
4*k1*n2 - 4*k1*n3
q3=---------------------
2
a33
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
2 2 2 2
4*b31 *k1*n2 - 4*b32 *k1*n3
q1=-------------------------------
2 2
a33 *b32
16 4 16 2 2
----*b31 *k1*n2 + ----*b31 *b32 *k1*n2
3 3
p56=----------------------------------------
4
a33 *b32
2 2
- 4*b31 *k1*m2 - 4*b32 *k1*m2
p55=--------------------------------
3
a33
2 3
4*b31 *b32*k1*m2 + 4*b32 *k1*m2
p54=---------------------------------
3
a33 *b31
2 2
- 4*b31 *k1*n3 - 4*b32 *k1*n3
p53=--------------------------------
3
a33
2 2
8*b31 *k1*n2 + 8*b32 *k1*n2
p52=-----------------------------
3
a33
3 2
- 8*b31 *k1*n2 - 8*b31*b32 *k1*n2
p51=------------------------------------
3
a33 *b32
40 4 80 2 2 40 4
- ----*b31 *k1*n2 - ----*b31 *b32 *k1*n2 - ----*b32 *k1*n2
3 3 3
p50=-------------------------------------------------------------
4
a33 *b32
p49=0
4 8 2 2 112 4
40*b31 *k1*n2 + ---*b31 *b32 *k1*n2 - -----*b32 *k1*n2
3 3
p48=--------------------------------------------------------
3 2
a33 *b32
p47=0
p46=0
16 4 32 2 2 16 4
- ----*b31 *k1*n2 - ----*b31 *b32 *k1*n2 - ----*b32 *k1*n2
3 3 3
p45=-------------------------------------------------------------
4
a33 *b32
5 8 3 2 112 4
40*b31 *k1*n2 + ---*b31 *b32 *k1*n2 - -----*b31*b32 *k1*n2
3 3
p44=------------------------------------------------------------
3 3
a33 *b32
p43=0
p42=0
16 2 16 2
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p41=-----------------------------------
2
a33 *b32
8*b32*k1*n3
p40=-------------
2
a33
8*b31*k1*n3
p39=-------------
2
a33
16 2 16 2
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p38=-----------------------------------
2
a33 *b32
p37=0
40 2 40 2
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p36=-----------------------------------
2
a33 *b32
p35=0
p34=0
p33=0
p32=0
3 2
- 8*b31 *k1*n2 - 8*b31*b32 *k1*n2
p31=------------------------------------
3
a33 *b32
p30=0
p29=0
3 2
8*b31 *k1*n2 + 8*b31*b32 *k1*n2
p28=---------------------------------
3
a33 *b32
2 2
- 8*b31 *k1*n2 - 8*b32 *k1*n2
p27=--------------------------------
3
a33
p26=0
112 2 160 2
-----*b31 *k1*n2 + -----*b32 *k1*n2
3 3
p25=-------------------------------------
2
a33 *b32
16*b31*k1*n2
p24=--------------
2
a33
- 4*k1*m2
p23=------------
a33
8*b32*k1*m2
p22=-------------
a33*b31
4*k1*m2
p21=---------
a33
p20=0
p19=0
2 2
8*b31 *k1*n2 + 8*b32 *k1*n2
p18=-----------------------------
3
a33
p17=0
p16=0
- 16*b31*k1*n2
p15=-----------------
2
a33
32 2 16 2
- ----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p14=--------------------------------------
2
a33 *b32
- 4*b32*k1*m2
p13=----------------
a33*b31
- 8*k1*m2
p12=------------
a33
4*b32*k1*m2
p11=-------------
a33*b31
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
- 4*k1*n2
p4=------------
a33
4*b31*k1*n2
p3=-------------
a33*b32
- 4*k1*n2
p2=------------
a33
4*b31*k1*n2
p1=-------------
a33*b32
k125=0
k124=0
k122=0
k121=0
k120=0
k119=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k118=------------------------------
3
a33
k117=0
k116=0
k115=0
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k114=------------------------------
3
a33
k113=0
k112=0
k110=0
k109=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k108=--------------------------
2
a33
k107=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k106=--------------------------
2
a33
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
2 2
8*b31 *k1 + 8*b32 *k1
k95=-----------------------
2
a33
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
2 2
8*b31 *k1 + 8*b32 *k1
k84=-----------------------
2
a33
k83=0
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
k64=0
k63=0
k62=0
k61=0
k60=0
k59=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k58=--------------------------
2
a33
k57=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k56=--------------------------
2
a33
k55=0
k54=0
k53=0
k52=0
k51=0
k50=0
k49=0
k48=0
k47=0
k46=0
k45=0
k44=0
k43=0
- 4*b32*k1
k42=-------------
a33
- 8*b31*k1
k41=-------------
a33
4*b32*k1
k40=----------
a33
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k30=0
k29=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k28=--------------------------
2
a33
k27=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k26=--------------------------
2
a33
k25=0
k24=0
k23=0
4*b31*k1
k22=----------
a33
- 8*b32*k1
k21=-------------
a33
- 4*b31*k1
k20=-------------
a33
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=0
k11=0
k10=0
k9=0
k8=0
k7=0
k6=0
k5=k1
k4=0
k3=2*k1
k2=0
Parameters
Apart from the condition that they must not vanish to give
a non-trivial solution and a non-singular solution with
non-vanishing denominators, the following parameters are free:
b32,n3,m2,k1,b31,n2,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(3*a33 *b31*b32*u1 + 3*a33 *b31*b32*u2 + 6*a33 *b31*b32*u3
2 2 2
+ 6*a33*b31 *b32*u3*v1 - 6*a33*b31 *n2*u1 + 6*a33*b31*b32 *u3*v2
+ 6*a33*b31*b32*m2*v2 + 6*a33*b31*b32*n2*u2 + 6*a33*b31*b32*n3*u3
2 3 2
- 6*a33*b32 *m2*v1 + 20*b31 *n2*v3 + 8*b31*b32 *n2*v3)/(6*a33*b31*b32)
4 2 3 4 4 2 3 2 2 4 2 3 4
INT=(k1*(9*a33 *b31 *b32 *u1 + 18*a33 *b31 *b32 *u1 *u2 + 9*a33 *b31 *b32 *u2
3 3 3 2 3 3 3
- 36*a33 *b31 *b32 *u1 *u3*v1 - 72*a33 *b31 *b32 *u1*u2*u3*v2
3 3 3 2 3 3 2 3
+ 36*a33 *b31 *b32 *u2 *u3*v1 + 36*a33 *b31 *b32 *n2*u1
3 3 2 2 3 2 4 2
+ 36*a33 *b31 *b32 *n2*u1*u2 + 36*a33 *b31 *b32 *u1 *u3*v2
3 2 4 3 2 4 2
- 72*a33 *b31 *b32 *u1*u2*u3*v1 - 36*a33 *b31 *b32 *u2 *u3*v2
3 2 3 2 3 2 3
+ 36*a33 *b31 *b32 *m2*u1 *v2 - 72*a33 *b31 *b32 *m2*u1*u2*v1
3 2 3 2 3 2 3 2
- 36*a33 *b31 *b32 *m2*u2 *v2 - 36*a33 *b31 *b32 *n2*u1 *u2
3 2 3 3 3 2 3 2
- 36*a33 *b31 *b32 *n2*u2 - 36*a33 *b31 *b32 *n3*u1 *u3
3 2 3 2 3 4 2
- 36*a33 *b31 *b32 *n3*u2 *u3 + 36*a33 *b31*b32 *m2*u1 *v1
3 4 3 4 2
+ 72*a33 *b31*b32 *m2*u1*u2*v2 - 36*a33 *b31*b32 *m2*u2 *v1
2 4 3 2 2 2 4 3 2 2
- 18*a33 *b31 *b32 *u1 *v1 - 18*a33 *b31 *b32 *u1 *v2
2 4 3 2 2 2 4 3
- 18*a33 *b31 *b32 *u1 *v3 + 72*a33 *b31 *b32 *u1*u3*v1*v3
2 4 3 2 2 2 4 3 2 2
- 18*a33 *b31 *b32 *u2 *v1 - 18*a33 *b31 *b32 *u2 *v2
2 4 3 2 2 2 4 3
- 18*a33 *b31 *b32 *u2 *v3 + 72*a33 *b31 *b32 *u2*u3*v2*v3
2 4 2 2 2 4 2
+ 120*a33 *b31 *b32 *n2*u1 *v3 - 96*a33 *b31 *b32 *n2*u1*u3*v1
2 4 2 2 2 4 2
+ 48*a33 *b31 *b32 *n2*u2 *v3 + 336*a33 *b31 *b32 *n2*u2*u3*v2
2 4 2 2 2 4 2 2
+ 48*a33 *b31 *b32 *n2*u3 *v3 + 36*a33 *b31 *b32*n2 *u1
2 3 3 2 3 3
+ 144*a33 *b31 *b32 *n2*u1*u3*v2 - 144*a33 *b31 *b32 *n2*u2*u3*v1
2 3 3 2 3 2
+ 72*a33 *b31 *b32 *n3*u1*u3*v3 + 144*a33 *b31 *b32 *m2*n2*u1*v2
2 3 2 2 2 3 2
- 72*a33 *b31 *b32 *n2 *u1*u2 - 72*a33 *b31 *b32 *n2*n3*u1*u3
2 2 5 2 2 2 2 5 2 2
- 18*a33 *b31 *b32 *u1 *v1 - 18*a33 *b31 *b32 *u1 *v2
2 2 5 2 2 2 2 5
- 18*a33 *b31 *b32 *u1 *v3 + 72*a33 *b31 *b32 *u1*u3*v1*v3
2 2 5 2 2 2 2 5 2 2
- 18*a33 *b31 *b32 *u2 *v1 - 18*a33 *b31 *b32 *u2 *v2
2 2 5 2 2 2 2 5
- 18*a33 *b31 *b32 *u2 *v3 + 72*a33 *b31 *b32 *u2*u3*v2*v3
2 2 4 2 2 2 4
+ 120*a33 *b31 *b32 *n2*u1 *v3 + 48*a33 *b31 *b32 *n2*u1*u3*v1
2 2 4 2 2 2 4
+ 48*a33 *b31 *b32 *n2*u2 *v3 + 480*a33 *b31 *b32 *n2*u2*u3*v2
2 2 4 2 2 2 4
+ 48*a33 *b31 *b32 *n2*u3 *v3 + 72*a33 *b31 *b32 *n3*u2*u3*v3
2 2 3 2 2 2 2 3 2 2
+ 36*a33 *b31 *b32 *m2 *v1 + 36*a33 *b31 *b32 *m2 *v2
2 2 3 2 2 3
+ 144*a33 *b31 *b32 *m2*n2*u1*v1 + 144*a33 *b31 *b32 *m2*n2*u2*v2
2 2 3 2 2 3 2 2
+ 72*a33 *b31 *b32 *m2*n3*u2*v3 + 36*a33 *b31 *b32 *n2 *u2
2 2 3 2 2 3 2 2
+ 72*a33 *b31 *b32 *n2*n3*u2*u3 - 36*a33 *b31 *b32 *n3 *u1
2 2 3 2 2 2 4
- 36*a33 *b31 *b32 *n3 *u2 + 144*a33 *b31*b32 *m2*n2*u2*v1
2 4 2 5 2 2
- 72*a33 *b31*b32 *m2*n3*u1*v3 + 36*a33 *b32 *m2 *v1
2 5 2 2 7
+ 36*a33 *b32 *m2 *v2 + 360*a33*b31 *n2*u3*v1*v3
6 6
+ 360*a33*b31 *b32*n2*u3*v2*v3 + 360*a33*b31 *m2*n2*v2*v3
5 3 2 5 2 2
- 36*a33*b31 *b32 *u3*v1*v3 - 72*a33*b31 *b32 *n2*u1*v2
5 2 2 5 2
- 72*a33*b31 *b32 *n2*u1*v3 + 72*a33*b31 *b32 *n2*u2*v1*v2
5 2 5
+ 24*a33*b31 *b32 *n2*u3*v1*v3 - 360*a33*b31 *b32*m2*n2*v1*v3
5 2 5 2
+ 480*a33*b31 *b32*n2 *u1*v3 + 144*a33*b31 *b32*n2 *u3*v1
4 4 2 4 3 2
- 36*a33*b31 *b32 *u3*v2*v3 - 36*a33*b31 *b32 *m2*v2*v3
4 3 4 3 2
- 72*a33*b31 *b32 *n2*u1*v1*v2 + 72*a33*b31 *b32 *n2*u2*v1
4 3 2 4 3
+ 72*a33*b31 *b32 *n2*u2*v3 + 24*a33*b31 *b32 *n2*u3*v2*v3
4 3 2 4 2
- 36*a33*b31 *b32 *n3*u3*v3 + 24*a33*b31 *b32 *m2*n2*v2*v3
4 2 2 4 2 2
- 192*a33*b31 *b32 *n2 *u2*v3 + 144*a33*b31 *b32 *n2 *u3*v2
4 2 4 2
+ 168*a33*b31 *b32 *n2*n3*u1*v1 - 48*a33*b31 *b32 *n2*n3*u2*v2
3 5 2 3 4 2
- 36*a33*b31 *b32 *u3*v1*v3 + 36*a33*b31 *b32 *m2*v1*v3
3 4 2 3 4 2
- 72*a33*b31 *b32 *n2*u1*v2 - 72*a33*b31 *b32 *n2*u1*v3
3 4 3 4
+ 72*a33*b31 *b32 *n2*u2*v1*v2 - 336*a33*b31 *b32 *n2*u3*v1*v3
3 3 3 3 2
- 24*a33*b31 *b32 *m2*n2*v1*v3 + 480*a33*b31 *b32 *n2 *u1*v3
3 3 2 3 3
+ 288*a33*b31 *b32 *n2 *u3*v1 - 72*a33*b31 *b32 *n2*n3*u1*v2
3 3 3 2 2
+ 72*a33*b31 *b32 *n2*n3*u2*v1 + 144*a33*b31 *b32 *m2*n2 *v1
2 6 2 2 5 2
- 36*a33*b31 *b32 *u3*v2*v3 - 36*a33*b31 *b32 *m2*v2*v3
2 5 2 5 2
- 72*a33*b31 *b32 *n2*u1*v1*v2 + 72*a33*b31 *b32 *n2*u2*v1
2 5 2 2 5
+ 72*a33*b31 *b32 *n2*u2*v3 - 336*a33*b31 *b32 *n2*u3*v2*v3
2 5 2 2 4
- 36*a33*b31 *b32 *n3*u3*v3 - 336*a33*b31 *b32 *m2*n2*v2*v3
2 4 2 2 4
- 192*a33*b31 *b32 *n2 *u2*v3 + 240*a33*b31 *b32 *n2*n3*u1*v1
2 4 2 3 2
+ 24*a33*b31 *b32 *n2*n3*u2*v2 - 144*a33*b31 *b32 *m2*n2 *v2
2 3 6 2
- 144*a33*b31 *b32 *m2*n2*n3*v3 + 36*a33*b31*b32 *m2*v1*v3
5 7 2
+ 336*a33*b31*b32 *m2*n2*v1*v3 + 360*b31 *n2 *v1*v2
6 2 2 6 2 2
- 48*b31 *b32 *n2*v1 *v3 - 120*b31 *b32 *n2*v2 *v3
6 2 3 6 2 2 6 2 2
+ 48*b31 *b32 *n2*v3 - 256*b31 *b32*n2 *v1 - 580*b31 *b32*n2 *v2
5 2 2 5 2
+ 576*b31 *b32 *n2 *v1*v2 + 312*b31 *b32*n2 *n3*v1
4 4 2 4 4 2
- 96*b31 *b32 *n2*v1 *v3 - 240*b31 *b32 *n2*v2 *v3
4 4 3 4 3 2 2 4 3 2 2
+ 48*b31 *b32 *n2*v3 - 200*b31 *b32 *n2 *v1 - 704*b31 *b32 *n2 *v2
4 2 2 4 2 2
- 312*b31 *b32 *n2 *n3*v2 - 312*b31 *b32 *n2*n3 *v3
3 4 2 3 3 2
+ 216*b31 *b32 *n2 *v1*v2 + 168*b31 *b32 *n2 *n3*v1
2 6 2 2 6 2
- 48*b31 *b32 *n2*v1 *v3 - 120*b31 *b32 *n2*v2 *v3
2 5 2 2 2 5 2 2
+ 56*b31 *b32 *n2 *v1 - 124*b31 *b32 *n2 *v2
2 4 2 2 4 2 4 2
- 168*b31 *b32 *n2 *n3*v2 - 168*b31 *b32 *n2*n3 *v3))/(9*a33 *b31
3
*b32 )
And again in machine readable form:
HAM=(3*a33**2*b31*b32*u1**2 + 3*a33**2*b31*b32*u2**2 + 6*a33**2*b31*b32*u3**2 +
6*a33*b31**2*b32*u3*v1 - 6*a33*b31**2*n2*u1 + 6*a33*b31*b32**2*u3*v2 + 6*a33*b31
*b32*m2*v2 + 6*a33*b31*b32*n2*u2 + 6*a33*b31*b32*n3*u3 - 6*a33*b32**2*m2*v1 + 20
*b31**3*n2*v3 + 8*b31*b32**2*n2*v3)/(6*a33*b31*b32)$
INT=(k1*(9*a33**4*b31**2*b32**3*u1**4 + 18*a33**4*b31**2*b32**3*u1**2*u2**2 + 9*
a33**4*b31**2*b32**3*u2**4 - 36*a33**3*b31**3*b32**3*u1**2*u3*v1 - 72*a33**3*b31
**3*b32**3*u1*u2*u3*v2 + 36*a33**3*b31**3*b32**3*u2**2*u3*v1 + 36*a33**3*b31**3*
b32**2*n2*u1**3 + 36*a33**3*b31**3*b32**2*n2*u1*u2**2 + 36*a33**3*b31**2*b32**4*
u1**2*u3*v2 - 72*a33**3*b31**2*b32**4*u1*u2*u3*v1 - 36*a33**3*b31**2*b32**4*u2**
2*u3*v2 + 36*a33**3*b31**2*b32**3*m2*u1**2*v2 - 72*a33**3*b31**2*b32**3*m2*u1*u2
*v1 - 36*a33**3*b31**2*b32**3*m2*u2**2*v2 - 36*a33**3*b31**2*b32**3*n2*u1**2*u2
- 36*a33**3*b31**2*b32**3*n2*u2**3 - 36*a33**3*b31**2*b32**3*n3*u1**2*u3 - 36*
a33**3*b31**2*b32**3*n3*u2**2*u3 + 36*a33**3*b31*b32**4*m2*u1**2*v1 + 72*a33**3*
b31*b32**4*m2*u1*u2*v2 - 36*a33**3*b31*b32**4*m2*u2**2*v1 - 18*a33**2*b31**4*b32
**3*u1**2*v1**2 - 18*a33**2*b31**4*b32**3*u1**2*v2**2 - 18*a33**2*b31**4*b32**3*
u1**2*v3**2 + 72*a33**2*b31**4*b32**3*u1*u3*v1*v3 - 18*a33**2*b31**4*b32**3*u2**
2*v1**2 - 18*a33**2*b31**4*b32**3*u2**2*v2**2 - 18*a33**2*b31**4*b32**3*u2**2*v3
**2 + 72*a33**2*b31**4*b32**3*u2*u3*v2*v3 + 120*a33**2*b31**4*b32**2*n2*u1**2*v3
- 96*a33**2*b31**4*b32**2*n2*u1*u3*v1 + 48*a33**2*b31**4*b32**2*n2*u2**2*v3 +
336*a33**2*b31**4*b32**2*n2*u2*u3*v2 + 48*a33**2*b31**4*b32**2*n2*u3**2*v3 + 36*
a33**2*b31**4*b32*n2**2*u1**2 + 144*a33**2*b31**3*b32**3*n2*u1*u3*v2 - 144*a33**
2*b31**3*b32**3*n2*u2*u3*v1 + 72*a33**2*b31**3*b32**3*n3*u1*u3*v3 + 144*a33**2*
b31**3*b32**2*m2*n2*u1*v2 - 72*a33**2*b31**3*b32**2*n2**2*u1*u2 - 72*a33**2*b31
**3*b32**2*n2*n3*u1*u3 - 18*a33**2*b31**2*b32**5*u1**2*v1**2 - 18*a33**2*b31**2*
b32**5*u1**2*v2**2 - 18*a33**2*b31**2*b32**5*u1**2*v3**2 + 72*a33**2*b31**2*b32
**5*u1*u3*v1*v3 - 18*a33**2*b31**2*b32**5*u2**2*v1**2 - 18*a33**2*b31**2*b32**5*
u2**2*v2**2 - 18*a33**2*b31**2*b32**5*u2**2*v3**2 + 72*a33**2*b31**2*b32**5*u2*
u3*v2*v3 + 120*a33**2*b31**2*b32**4*n2*u1**2*v3 + 48*a33**2*b31**2*b32**4*n2*u1*
u3*v1 + 48*a33**2*b31**2*b32**4*n2*u2**2*v3 + 480*a33**2*b31**2*b32**4*n2*u2*u3*
v2 + 48*a33**2*b31**2*b32**4*n2*u3**2*v3 + 72*a33**2*b31**2*b32**4*n3*u2*u3*v3 +
36*a33**2*b31**2*b32**3*m2**2*v1**2 + 36*a33**2*b31**2*b32**3*m2**2*v2**2 + 144
*a33**2*b31**2*b32**3*m2*n2*u1*v1 + 144*a33**2*b31**2*b32**3*m2*n2*u2*v2 + 72*
a33**2*b31**2*b32**3*m2*n3*u2*v3 + 36*a33**2*b31**2*b32**3*n2**2*u2**2 + 72*a33
**2*b31**2*b32**3*n2*n3*u2*u3 - 36*a33**2*b31**2*b32**3*n3**2*u1**2 - 36*a33**2*
b31**2*b32**3*n3**2*u2**2 + 144*a33**2*b31*b32**4*m2*n2*u2*v1 - 72*a33**2*b31*
b32**4*m2*n3*u1*v3 + 36*a33**2*b32**5*m2**2*v1**2 + 36*a33**2*b32**5*m2**2*v2**2
+ 360*a33*b31**7*n2*u3*v1*v3 + 360*a33*b31**6*b32*n2*u3*v2*v3 + 360*a33*b31**6*
m2*n2*v2*v3 - 36*a33*b31**5*b32**3*u3*v1*v3**2 - 72*a33*b31**5*b32**2*n2*u1*v2**
2 - 72*a33*b31**5*b32**2*n2*u1*v3**2 + 72*a33*b31**5*b32**2*n2*u2*v1*v2 + 24*a33
*b31**5*b32**2*n2*u3*v1*v3 - 360*a33*b31**5*b32*m2*n2*v1*v3 + 480*a33*b31**5*b32
*n2**2*u1*v3 + 144*a33*b31**5*b32*n2**2*u3*v1 - 36*a33*b31**4*b32**4*u3*v2*v3**2
- 36*a33*b31**4*b32**3*m2*v2*v3**2 - 72*a33*b31**4*b32**3*n2*u1*v1*v2 + 72*a33*
b31**4*b32**3*n2*u2*v1**2 + 72*a33*b31**4*b32**3*n2*u2*v3**2 + 24*a33*b31**4*b32
**3*n2*u3*v2*v3 - 36*a33*b31**4*b32**3*n3*u3*v3**2 + 24*a33*b31**4*b32**2*m2*n2*
v2*v3 - 192*a33*b31**4*b32**2*n2**2*u2*v3 + 144*a33*b31**4*b32**2*n2**2*u3*v2 +
168*a33*b31**4*b32**2*n2*n3*u1*v1 - 48*a33*b31**4*b32**2*n2*n3*u2*v2 - 36*a33*
b31**3*b32**5*u3*v1*v3**2 + 36*a33*b31**3*b32**4*m2*v1*v3**2 - 72*a33*b31**3*b32
**4*n2*u1*v2**2 - 72*a33*b31**3*b32**4*n2*u1*v3**2 + 72*a33*b31**3*b32**4*n2*u2*
v1*v2 - 336*a33*b31**3*b32**4*n2*u3*v1*v3 - 24*a33*b31**3*b32**3*m2*n2*v1*v3 +
480*a33*b31**3*b32**3*n2**2*u1*v3 + 288*a33*b31**3*b32**3*n2**2*u3*v1 - 72*a33*
b31**3*b32**3*n2*n3*u1*v2 + 72*a33*b31**3*b32**3*n2*n3*u2*v1 + 144*a33*b31**3*
b32**2*m2*n2**2*v1 - 36*a33*b31**2*b32**6*u3*v2*v3**2 - 36*a33*b31**2*b32**5*m2*
v2*v3**2 - 72*a33*b31**2*b32**5*n2*u1*v1*v2 + 72*a33*b31**2*b32**5*n2*u2*v1**2 +
72*a33*b31**2*b32**5*n2*u2*v3**2 - 336*a33*b31**2*b32**5*n2*u3*v2*v3 - 36*a33*
b31**2*b32**5*n3*u3*v3**2 - 336*a33*b31**2*b32**4*m2*n2*v2*v3 - 192*a33*b31**2*
b32**4*n2**2*u2*v3 + 240*a33*b31**2*b32**4*n2*n3*u1*v1 + 24*a33*b31**2*b32**4*n2
*n3*u2*v2 - 144*a33*b31**2*b32**3*m2*n2**2*v2 - 144*a33*b31**2*b32**3*m2*n2*n3*
v3 + 36*a33*b31*b32**6*m2*v1*v3**2 + 336*a33*b31*b32**5*m2*n2*v1*v3 + 360*b31**7
*n2**2*v1*v2 - 48*b31**6*b32**2*n2*v1**2*v3 - 120*b31**6*b32**2*n2*v2**2*v3 + 48
*b31**6*b32**2*n2*v3**3 - 256*b31**6*b32*n2**2*v1**2 - 580*b31**6*b32*n2**2*v2**
2 + 576*b31**5*b32**2*n2**2*v1*v2 + 312*b31**5*b32*n2**2*n3*v1 - 96*b31**4*b32**
4*n2*v1**2*v3 - 240*b31**4*b32**4*n2*v2**2*v3 + 48*b31**4*b32**4*n2*v3**3 - 200*
b31**4*b32**3*n2**2*v1**2 - 704*b31**4*b32**3*n2**2*v2**2 - 312*b31**4*b32**2*n2
**2*n3*v2 - 312*b31**4*b32**2*n2*n3**2*v3 + 216*b31**3*b32**4*n2**2*v1*v2 + 168*
b31**3*b32**3*n2**2*n3*v1 - 48*b31**2*b32**6*n2*v1**2*v3 - 120*b31**2*b32**6*n2*
v2**2*v3 + 56*b31**2*b32**5*n2**2*v1**2 - 124*b31**2*b32**5*n2**2*v2**2 - 168*
b31**2*b32**4*n2**2*n3*v2 - 168*b31**2*b32**4*n2*n3**2*v3))/(9*a33**4*b31**2*b32
**3)$