Solution 7 to problem e3null
Expressions |
Parameters |
Relevance |
Back to problem e3null
Expressions
The solution is given through the following expressions:
a22=3*a33
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
1 2
- ----*b33
12
c33=--------------
a33
1
- ---*b33*n1
3
m1=---------------
a33
1
- ---*b33*n2
3
m2=---------------
a33
1
- ---*b33*n3
2
m3=---------------
a33
1 2 1 2 1 2
- ---*a33 *b33*n3*q1 + ----*b33*k1*n1 *n3 - ----*b33*k10*n1 *n3
6 24 24
r6=------------------------------------------------------------------
4
a33
1 2 1 2 1 2
r5=( - ----*a33 *b33*n2*q1 + ----*b33*k1*n1 *n2 + ----*b33*k1*n2*n3
12 48 48
1 2 1 2 4
- ----*b33*k10*n1 *n2 - ----*b33*k10*n2*n3 )/a33
48 48
1 2 1 3 1 2
r4=( - ----*a33 *b33*n1*q1 + ----*b33*k1*n1 + ----*b33*k1*n1*n3
12 48 48
1 3 1 2 4
- ----*b33*k10*n1 - ----*b33*k10*n1*n3 )/a33
48 48
1 2 1 2 1 3 1 2 1 3
---*a33 *n3*q1 - ---*k1*n1 *n3 + ---*k1*n3 + ---*k10*n1 *n3 - ---*k10*n3
2 8 8 8 8
r3=----------------------------------------------------------------------------
3
a33
1 2 1 2 1 2 1 2
r2=(---*a33 *n2*q1 - ---*k1*n1 *n2 + ---*k1*n2*n3 + ---*k10*n1 *n2
2 8 8 8
1 2 3
- ---*k10*n2*n3 )/a33
8
1 2 1 3 1 2 1 3 1 2
---*a33 *n1*q1 - ---*k1*n1 + ---*k1*n1*n3 + ---*k10*n1 - ---*k10*n1*n3
2 8 8 8 8
r1=----------------------------------------------------------------------------
3
a33
1 2 1 2
----*b33 *k1*n2*n3 + ----*b33 *k10*n2*n3
36 72
q19=------------------------------------------
4
a33
1 2 1 2
----*b33 *k1*n1*n3 + ----*b33 *k10*n1*n3
36 72
q18=------------------------------------------
4
a33
1 2 1 2 1 2 1 2
---*a33 *b33*q1 - ---*b33*k1*n3 + ----*b33*k10*n1 + ----*b33*k10*n3
3 6 12 12
q17=------------------------------------------------------------------------
3
a33
1 1
- ---*b33*k1*n2*n3 + ----*b33*k10*n2*n3
6 12
q16=------------------------------------------
3
a33
1 1
- ---*b33*k1*n1*n3 + ----*b33*k10*n1*n3
6 12
q15=------------------------------------------
3
a33
1 2 2 1 2 2 1 2 2
q14=(----*a33 *b33 *q1 - -----*b33 *k1*n1 + -----*b33 *k1*n2
36 144 144
1 2 2 1 2 2 1 2 2 4
- ----*b33 *k1*n3 + -----*b33 *k10*n1 + -----*b33 *k10*n2 )/a33
36 144 144
1 2 1 2
----*b33 *k1*n1*n2 + ----*b33 *k10*n1*n2
72 72
q13=------------------------------------------
4
a33
1
- ----*b33*k1*n2*n3
12
q12=----------------------
3
a33
1 2 1 2
----*b33*k1*n1 - ----*b33*k1*n2
12 12
q11=-----------------------------------
3
a33
1
- ----*b33*k1*n1*n2
12
q10=----------------------
3
a33
1 2 2 1 2 2 1 2 2
----*a33 *b33 *q1 - ----*b33 *k1*n3 + ----*b33 *k10*n1
36 36 72
q9=----------------------------------------------------------
4
a33
1
- ----*b33*k1*n1*n3
12
q8=----------------------
3
a33
1
- ----*b33*k1*n1*n2
12
q7=----------------------
3
a33
1 1
---*k1*n2*n3 - ---*k10*n2*n3
2 2
q5=------------------------------
2
a33
1 1
---*k1*n1*n3 - ---*k10*n1*n3
2 2
q4=------------------------------
2
a33
2 1 2 1 2 1 2 1 2
a33 *q1 - ---*k1*n1 + ---*k1*n2 + ---*k10*n1 - ---*k10*n2
4 4 4 4
q3=---------------------------------------------------------------
2
a33
1 1
---*k1*n1*n2 - ---*k10*n1*n2
2 2
q2=------------------------------
2
a33
1 2 1 3
- ---*a33 *b33*k20*n3 + -----*b33 *k1*n3
6 108
p50=-------------------------------------------
4
a33
1 2 1 3 1 3
- ----*a33 *b33*k20*n2 + -----*b33 *k1*n2 + -----*b33 *k10*n2
12 216 432
p49=----------------------------------------------------------------
4
a33
1 2 1 3 1 3
- ----*a33 *b33*k20*n1 + -----*b33 *k1*n1 + -----*b33 *k10*n1
12 216 432
p48=----------------------------------------------------------------
4
a33
1 2 5 2 1 2
---*a33 *k20*n3 - ----*b33 *k1*n3 + ----*b33 *k10*n3
2 36 72
p47=------------------------------------------------------
3
a33
1 2 1 2 1 2
---*a33 *k20*n2 - ----*b33 *k1*n2 + ----*b33 *k10*n2
2 36 72
p46=------------------------------------------------------
3
a33
1 2 1 2 1 2
---*a33 *k20*n1 - ----*b33 *k1*n1 + ----*b33 *k10*n1
2 36 72
p45=------------------------------------------------------
3
a33
1
- ---*b33*k20*n3
6
p44=-------------------
2
a33
p43=0
1 2 1 2
- ----*b33 *k1*n2 - ----*b33 *k10*n2
18 36
p42=---------------------------------------
3
a33
p41=0
p40=0
1
- ---*b33*k20*n3
6
p39=-------------------
2
a33
1 2 1 2 1 2
- ---*a33 *k20*n1 - ----*b33 *k1*n1 - ----*b33 *k10*n1
2 18 36
p38=---------------------------------------------------------
3
a33
p37=0
1 1
---*b33*k1*n3 - ---*b33*k10*n3
3 3
p36=--------------------------------
2
a33
1 1
---*b33*k1*n2 - ---*b33*k10*n2
3 6
p35=--------------------------------
2
a33
1 1
---*b33*k1*n1 - ----*b33*k10*n1
2 12
p34=---------------------------------
2
a33
1
- ---*b33*k1*n3
3
p33=------------------
2
a33
p32=0
1
- ---*b33*k1*n3
3
p31=------------------
2
a33
1
- ----*b33*k20*n2
12
p30=--------------------
2
a33
1
- ----*b33*k20*n1
12
p29=--------------------
2
a33
1
---*k20*n3
2
p28=------------
a33
1
---*k20*n2
2
p27=------------
a33
1
---*k20*n1
2
p26=------------
a33
1
- ----*b33*k20*n2
12
p25=--------------------
2
a33
p24=0
1
- ---*k20*n1
2
p23=---------------
a33
1
- ----*b33*k10*n2
12
p22=--------------------
2
a33
p21=0
p20=0
1 1
- ---*b33*k1*n2 - ----*b33*k10*n2
6 12
p19=------------------------------------
2
a33
1 1
---*b33*k1*n1 + ----*b33*k10*n1
6 12
p18=---------------------------------
2
a33
1 1
- ---*b33*k1*n2 - ----*b33*k10*n2
6 12
p17=------------------------------------
2
a33
1
- ----*b33*k20*n1
12
p16=--------------------
2
a33
1
---*k20*n3
2
p15=------------
a33
1
---*k20*n2
2
p14=------------
a33
1
- ----*b33*k10*n1
12
p13=--------------------
2
a33
p12=0
1 1
- ---*b33*k1*n1 - ----*b33*k10*n1
6 12
p11=------------------------------------
2
a33
1
---*k10*n3
2
p10=------------
a33
1
---*k10*n2
2
p9=------------
a33
1
---*k10*n1
2
p8=------------
a33
1
k1*n3 - ---*k10*n3
2
p7=--------------------
a33
p6=0
1
k1*n3 - ---*k10*n3
2
p5=--------------------
a33
1
k1*n2 - ---*k10*n2
2
p4=--------------------
a33
1
k1*n1 - ---*k10*n1
2
p3=--------------------
a33
1
k1*n2 - ---*k10*n2
2
p2=--------------------
a33
1
k1*n1 - ---*k10*n1
2
p1=--------------------
a33
k104=0
k103=0
1 2 1 3
---*a33 *b33*k20 - ----*b33 *k1
3 54
k102=---------------------------------
3
a33
k101=0
k100=0
1 2 2 1 4
----*a33 *b33 *k20 - -----*b33 *k1
36 648
k99=------------------------------------
4
a33
k98=0
k97=0
k96=0
k95=0
1 2 2 1 4
----*a33 *b33 *k20 - -----*b33 *k1
36 648
k94=------------------------------------
4
a33
k93=0
k92=0
2 1 2 1 2
a33 *k20 + ---*b33 *k1 - ----*b33 *k10
9 36
k91=----------------------------------------
2
a33
k90=0
k89=0
2 1 2
a33 *k20 - ----*b33 *k1
18
k88=-------------------------
2
a33
k87=0
2 1 2
a33 *k20 - ----*b33 *k1
18
k86=-------------------------
2
a33
k85=0
k84=0
1
---*b33*k20
3
k83=-------------
a33
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
k76=2*k20
k75=0
k74=0
k73=0
k72=0
k71=0
1
---*b33*k20
3
k70=-------------
a33
k69=0
k68=0
k67=0
k66=0
1
---*b33*k10
3
k65=-------------
a33
k64=0
k63=0
2
---*b33*k1
3
k62=------------
a33
k61=0
2
---*b33*k1
3
k60=------------
a33
k59=0
k58=0
k57=0
k56=0
1 2 2 1 4
----*a33 *b33 *k20 - ------*b33 *k1
36 1296
k55=-------------------------------------
4
a33
k54=0
k53=0
k52=0
k51=0
1 2 2 1 4
----*a33 *b33 *k20 - -----*b33 *k1
18 648
k50=------------------------------------
4
a33
k49=0
k48=0
k47=0
k46=0
k45=0
k44=2*k20
k43=0
k42=k20
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k31=0
k30=0
k29=0
k28=0
k27=0
k26=0
1 2 2 1 4
----*a33 *b33 *k20 - ------*b33 *k1
36 1296
k25=-------------------------------------
4
a33
k24=0
k23=0
k21=0
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=k10
k11=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:
q1,k20,k10,k1,b33,n1,n3,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=(36*a33 *u1 + 36*a33 *u2 + 12*a33 *u3 + 12*a33*b33*u3*v3 + 12*a33*n1*u1
2 2
+ 12*a33*n2*u2 + 12*a33*n3*u3 - b33 *v3 - 4*b33*n1*v1 - 4*b33*n2*v2
- 6*b33*n3*v3)/(12*a33)
4 4 4 2 2 4 4
INT=(1296*a33 *k1*u1 + 2592*a33 *k1*u1 *u2 + 1296*a33 *k1*u2
4 2 2 4 2 2 4 2 2
+ 1296*a33 *k10*u1 *u3 + 1296*a33 *k10*u2 *u3 + 1296*a33 *k20*u1 *v2
4 2 2 4 2 2 4 2 2
+ 1296*a33 *k20*u1 *v3 + 1296*a33 *k20*u2 *v1 + 2592*a33 *k20*u2 *v2
4 2 2 4
+ 1296*a33 *k20*u2 *v3 + 2592*a33 *k20*u2*u3*v2*v3
4 2 2 4 2 4 2
+ 1296*a33 *k20*u3 *v3 + 1296*a33 *q1*u1 + 1296*a33 *q1*u2
3 2 3 2
+ 864*a33 *b33*k1*u1 *u3*v3 + 864*a33 *b33*k1*u2 *u3*v3
3 3 3 2
+ 432*a33 *b33*k10*u3 *v3 + 432*a33 *b33*k20*u3*v1 *v3
3 2 3 3
+ 432*a33 *b33*k20*u3*v2 *v3 + 432*a33 *b33*k20*u3*v3
3 3 3 3 2
+ 432*a33 *b33*q1*u3*v3 + 1296*a33 *k1*n1*u1 + 1296*a33 *k1*n1*u1*u2
3 2 3 3 3 2
+ 1296*a33 *k1*n2*u1 *u2 + 1296*a33 *k1*n2*u2 + 1296*a33 *k1*n3*u1 *u3
3 2 3 3 3 2
+ 1296*a33 *k1*n3*u2 *u3 - 648*a33 *k10*n1*u1 - 648*a33 *k10*n1*u1*u2
3 2 3 2 3 3
+ 648*a33 *k10*n1*u1*u3 - 648*a33 *k10*n2*u1 *u2 - 648*a33 *k10*n2*u2
3 2 3 2 3 2
+ 648*a33 *k10*n2*u2*u3 - 648*a33 *k10*n3*u1 *u3 - 648*a33 *k10*n3*u2 *u3
3 3 3 2 3 2
+ 648*a33 *k10*n3*u3 + 648*a33 *k20*n1*u1*v2 + 648*a33 *k20*n1*u1*v3
3 3
- 648*a33 *k20*n1*u2*v1*v2 - 648*a33 *k20*n1*u3*v1*v3
3 2 3 2 3 2
+ 648*a33 *k20*n2*u2*v1 + 648*a33 *k20*n2*u2*v2 + 648*a33 *k20*n2*u2*v3
3 2 3 2 3 2
+ 648*a33 *k20*n3*u3*v1 + 648*a33 *k20*n3*u3*v2 + 648*a33 *k20*n3*u3*v3
3 3 3
+ 648*a33 *n1*q1*u1 + 648*a33 *n2*q1*u2 + 648*a33 *n3*q1*u3
2 2 2 2 2 2 2 2
- 72*a33 *b33 *k1*u1 *v3 - 72*a33 *b33 *k1*u2 *v3
2 2 2 2 2 2 2 2
+ 144*a33 *b33 *k1*u3 *v3 - 36*a33 *b33 *k10*u3 *v3
2 2 4 2 2 2 2
+ 36*a33 *b33 *k20*v1 + 72*a33 *b33 *k20*v1 *v2
2 2 2 2 2 2 4
+ 36*a33 *b33 *k20*v1 *v3 + 36*a33 *b33 *k20*v2
2 2 2 2 2 2 2 2 2 2
+ 36*a33 *b33 *k20*v2 *v3 + 36*a33 *b33 *q1*v1 + 36*a33 *b33 *q1*v2
2 2
+ 216*a33 *b33*k1*n1*u1*u2*v2 + 648*a33 *b33*k1*n1*u1*u3*v3
2 2 2 2
- 216*a33 *b33*k1*n1*u2 *v1 - 216*a33 *b33*k1*n2*u1 *v2
2 2 2
- 216*a33 *b33*k1*n2*u2 *v2 + 432*a33 *b33*k1*n2*u2*u3*v3
2 2 2 2
- 432*a33 *b33*k1*n3*u1 *v3 - 432*a33 *b33*k1*n3*u2 *v3
2 2 2
+ 432*a33 *b33*k1*n3*u3 *v3 + 108*a33 *b33*k10*n1*u1*u2*v2
2 2 2
- 108*a33 *b33*k10*n1*u1*u3*v3 - 108*a33 *b33*k10*n1*u2 *v1
2 2 2 2
- 108*a33 *b33*k10*n1*u3 *v1 - 108*a33 *b33*k10*n2*u1 *v2
2 2 2
- 108*a33 *b33*k10*n2*u2 *v2 - 216*a33 *b33*k10*n2*u2*u3*v3
2 2 2 2
- 108*a33 *b33*k10*n2*u3 *v2 - 432*a33 *b33*k10*n3*u3 *v3
2 3 2 2
- 108*a33 *b33*k20*n1*v1 - 108*a33 *b33*k20*n1*v1*v2
2 2 2 2
- 108*a33 *b33*k20*n1*v1*v3 - 108*a33 *b33*k20*n2*v1 *v2
2 3 2 2
- 108*a33 *b33*k20*n2*v2 - 108*a33 *b33*k20*n2*v2*v3
2 2 2 2
- 216*a33 *b33*k20*n3*v1 *v3 - 216*a33 *b33*k20*n3*v2 *v3
2 3 2 2
- 216*a33 *b33*k20*n3*v3 - 108*a33 *b33*n1*q1*v1 - 108*a33 *b33*n2*q1*v2
2 2 2 2 2
- 216*a33 *b33*n3*q1*v3 - 324*a33 *k1*n1 *u2 + 648*a33 *k1*n1*n2*u1*u2
2 2 2 2 2
+ 648*a33 *k1*n1*n3*u1*u3 + 324*a33 *k1*n2 *u2 + 648*a33 *k1*n2*n3*u2*u3
2 2 2 2
+ 324*a33 *k10*n1 *u2 - 648*a33 *k10*n1*n2*u1*u2
2 2 2 2
- 648*a33 *k10*n1*n3*u1*u3 - 324*a33 *k10*n2 *u2
2 3 3
- 648*a33 *k10*n2*n3*u2*u3 - 24*a33*b33 *k1*u3*v3
2 2 2
- 36*a33*b33 *k1*n1*u1*v3 - 72*a33*b33 *k1*n1*u3*v1*v3
2 2 2
- 36*a33*b33 *k1*n2*u2*v3 - 72*a33*b33 *k1*n2*u3*v2*v3
2 2 2 2
- 180*a33*b33 *k1*n3*u3*v3 + 18*a33*b33 *k10*n1*u1*v3
2 2 2
- 36*a33*b33 *k10*n1*u3*v1*v3 + 18*a33*b33 *k10*n2*u2*v3
2 2 2
- 36*a33*b33 *k10*n2*u3*v2*v3 + 18*a33*b33 *k10*n3*u3*v3
2
+ 108*a33*b33*k1*n1 *u2*v2 - 108*a33*b33*k1*n1*n2*u1*v2
- 108*a33*b33*k1*n1*n2*u2*v1 - 216*a33*b33*k1*n1*n3*u1*v3
2
- 108*a33*b33*k1*n1*n3*u3*v1 - 108*a33*b33*k1*n2 *u2*v2
- 216*a33*b33*k1*n2*n3*u2*v3 - 108*a33*b33*k1*n2*n3*u3*v2
2 2
- 216*a33*b33*k1*n3 *u3*v3 + 108*a33*b33*k10*n1 *u3*v3
+ 108*a33*b33*k10*n1*n3*u1*v3 + 108*a33*b33*k10*n2*n3*u2*v3
2 3 2
+ 108*a33*b33*k10*n3 *u3*v3 - 162*a33*k1*n1 *u1 - 162*a33*k1*n1 *n2*u2
2 2 2
- 162*a33*k1*n1 *n3*u3 + 162*a33*k1*n1*n3 *u1 + 162*a33*k1*n2*n3 *u2
3 3 2
+ 162*a33*k1*n3 *u3 + 162*a33*k10*n1 *u1 + 162*a33*k10*n1 *n2*u2
2 2 2
+ 162*a33*k10*n1 *n3*u3 - 162*a33*k10*n1*n3 *u1 - 162*a33*k10*n2*n3 *u2
3 4 4 4 2 2 4 2 2
- 162*a33*k10*n3 *u3 - b33 *k1*v1 - 2*b33 *k1*v1 *v2 - 2*b33 *k1*v1 *v3
4 4 4 2 2 3 2
- b33 *k1*v2 - 2*b33 *k1*v2 *v3 + 6*b33 *k1*n1*v1*v3
3 2 3 3 3 2
+ 6*b33 *k1*n2*v2*v3 + 12*b33 *k1*n3*v3 + 3*b33 *k10*n1*v1*v3
3 2 2 2 2 2
+ 3*b33 *k10*n2*v2*v3 - 9*b33 *k1*n1 *v2 + 18*b33 *k1*n1*n2*v1*v2
2 2 2 2 2
+ 36*b33 *k1*n1*n3*v1*v3 + 9*b33 *k1*n2 *v2 + 36*b33 *k1*n2*n3*v2*v3
2 2 2 2 2 2 2 2 2
- 36*b33 *k1*n3 *v1 - 36*b33 *k1*n3 *v2 + 18*b33 *k10*n1 *v1
2 2 2 2 2
+ 9*b33 *k10*n1 *v2 + 18*b33 *k10*n1*n2*v1*v2 + 18*b33 *k10*n1*n3*v1*v3
2 2 2 2 3
+ 9*b33 *k10*n2 *v2 + 18*b33 *k10*n2*n3*v2*v3 + 27*b33*k1*n1 *v1
2 2 2
+ 27*b33*k1*n1 *n2*v2 + 54*b33*k1*n1 *n3*v3 + 27*b33*k1*n1*n3 *v1
2 3 2
+ 27*b33*k1*n2*n3 *v2 - 27*b33*k10*n1 *v1 - 27*b33*k10*n1 *n2*v2
2 2 2
- 54*b33*k10*n1 *n3*v3 - 27*b33*k10*n1*n3 *v1 - 27*b33*k10*n2*n3 *v2)/(
4
1296*a33 )
And again in machine readable form:
HAM=(36*a33**2*u1**2 + 36*a33**2*u2**2 + 12*a33**2*u3**2 + 12*a33*b33*u3*v3 + 12
*a33*n1*u1 + 12*a33*n2*u2 + 12*a33*n3*u3 - b33**2*v3**2 - 4*b33*n1*v1 - 4*b33*n2
*v2 - 6*b33*n3*v3)/(12*a33)$
INT=(1296*a33**4*k1*u1**4 + 2592*a33**4*k1*u1**2*u2**2 + 1296*a33**4*k1*u2**4 +
1296*a33**4*k10*u1**2*u3**2 + 1296*a33**4*k10*u2**2*u3**2 + 1296*a33**4*k20*u1**
2*v2**2 + 1296*a33**4*k20*u1**2*v3**2 + 1296*a33**4*k20*u2**2*v1**2 + 2592*a33**
4*k20*u2**2*v2**2 + 1296*a33**4*k20*u2**2*v3**2 + 2592*a33**4*k20*u2*u3*v2*v3 +
1296*a33**4*k20*u3**2*v3**2 + 1296*a33**4*q1*u1**2 + 1296*a33**4*q1*u2**2 + 864*
a33**3*b33*k1*u1**2*u3*v3 + 864*a33**3*b33*k1*u2**2*u3*v3 + 432*a33**3*b33*k10*
u3**3*v3 + 432*a33**3*b33*k20*u3*v1**2*v3 + 432*a33**3*b33*k20*u3*v2**2*v3 + 432
*a33**3*b33*k20*u3*v3**3 + 432*a33**3*b33*q1*u3*v3 + 1296*a33**3*k1*n1*u1**3 +
1296*a33**3*k1*n1*u1*u2**2 + 1296*a33**3*k1*n2*u1**2*u2 + 1296*a33**3*k1*n2*u2**
3 + 1296*a33**3*k1*n3*u1**2*u3 + 1296*a33**3*k1*n3*u2**2*u3 - 648*a33**3*k10*n1*
u1**3 - 648*a33**3*k10*n1*u1*u2**2 + 648*a33**3*k10*n1*u1*u3**2 - 648*a33**3*k10
*n2*u1**2*u2 - 648*a33**3*k10*n2*u2**3 + 648*a33**3*k10*n2*u2*u3**2 - 648*a33**3
*k10*n3*u1**2*u3 - 648*a33**3*k10*n3*u2**2*u3 + 648*a33**3*k10*n3*u3**3 + 648*
a33**3*k20*n1*u1*v2**2 + 648*a33**3*k20*n1*u1*v3**2 - 648*a33**3*k20*n1*u2*v1*v2
- 648*a33**3*k20*n1*u3*v1*v3 + 648*a33**3*k20*n2*u2*v1**2 + 648*a33**3*k20*n2*
u2*v2**2 + 648*a33**3*k20*n2*u2*v3**2 + 648*a33**3*k20*n3*u3*v1**2 + 648*a33**3*
k20*n3*u3*v2**2 + 648*a33**3*k20*n3*u3*v3**2 + 648*a33**3*n1*q1*u1 + 648*a33**3*
n2*q1*u2 + 648*a33**3*n3*q1*u3 - 72*a33**2*b33**2*k1*u1**2*v3**2 - 72*a33**2*b33
**2*k1*u2**2*v3**2 + 144*a33**2*b33**2*k1*u3**2*v3**2 - 36*a33**2*b33**2*k10*u3
**2*v3**2 + 36*a33**2*b33**2*k20*v1**4 + 72*a33**2*b33**2*k20*v1**2*v2**2 + 36*
a33**2*b33**2*k20*v1**2*v3**2 + 36*a33**2*b33**2*k20*v2**4 + 36*a33**2*b33**2*
k20*v2**2*v3**2 + 36*a33**2*b33**2*q1*v1**2 + 36*a33**2*b33**2*q1*v2**2 + 216*
a33**2*b33*k1*n1*u1*u2*v2 + 648*a33**2*b33*k1*n1*u1*u3*v3 - 216*a33**2*b33*k1*n1
*u2**2*v1 - 216*a33**2*b33*k1*n2*u1**2*v2 - 216*a33**2*b33*k1*n2*u2**2*v2 + 432*
a33**2*b33*k1*n2*u2*u3*v3 - 432*a33**2*b33*k1*n3*u1**2*v3 - 432*a33**2*b33*k1*n3
*u2**2*v3 + 432*a33**2*b33*k1*n3*u3**2*v3 + 108*a33**2*b33*k10*n1*u1*u2*v2 - 108
*a33**2*b33*k10*n1*u1*u3*v3 - 108*a33**2*b33*k10*n1*u2**2*v1 - 108*a33**2*b33*
k10*n1*u3**2*v1 - 108*a33**2*b33*k10*n2*u1**2*v2 - 108*a33**2*b33*k10*n2*u2**2*
v2 - 216*a33**2*b33*k10*n2*u2*u3*v3 - 108*a33**2*b33*k10*n2*u3**2*v2 - 432*a33**
2*b33*k10*n3*u3**2*v3 - 108*a33**2*b33*k20*n1*v1**3 - 108*a33**2*b33*k20*n1*v1*
v2**2 - 108*a33**2*b33*k20*n1*v1*v3**2 - 108*a33**2*b33*k20*n2*v1**2*v2 - 108*
a33**2*b33*k20*n2*v2**3 - 108*a33**2*b33*k20*n2*v2*v3**2 - 216*a33**2*b33*k20*n3
*v1**2*v3 - 216*a33**2*b33*k20*n3*v2**2*v3 - 216*a33**2*b33*k20*n3*v3**3 - 108*
a33**2*b33*n1*q1*v1 - 108*a33**2*b33*n2*q1*v2 - 216*a33**2*b33*n3*q1*v3 - 324*
a33**2*k1*n1**2*u2**2 + 648*a33**2*k1*n1*n2*u1*u2 + 648*a33**2*k1*n1*n3*u1*u3 +
324*a33**2*k1*n2**2*u2**2 + 648*a33**2*k1*n2*n3*u2*u3 + 324*a33**2*k10*n1**2*u2
**2 - 648*a33**2*k10*n1*n2*u1*u2 - 648*a33**2*k10*n1*n3*u1*u3 - 324*a33**2*k10*
n2**2*u2**2 - 648*a33**2*k10*n2*n3*u2*u3 - 24*a33*b33**3*k1*u3*v3**3 - 36*a33*
b33**2*k1*n1*u1*v3**2 - 72*a33*b33**2*k1*n1*u3*v1*v3 - 36*a33*b33**2*k1*n2*u2*v3
**2 - 72*a33*b33**2*k1*n2*u3*v2*v3 - 180*a33*b33**2*k1*n3*u3*v3**2 + 18*a33*b33
**2*k10*n1*u1*v3**2 - 36*a33*b33**2*k10*n1*u3*v1*v3 + 18*a33*b33**2*k10*n2*u2*v3
**2 - 36*a33*b33**2*k10*n2*u3*v2*v3 + 18*a33*b33**2*k10*n3*u3*v3**2 + 108*a33*
b33*k1*n1**2*u2*v2 - 108*a33*b33*k1*n1*n2*u1*v2 - 108*a33*b33*k1*n1*n2*u2*v1 -
216*a33*b33*k1*n1*n3*u1*v3 - 108*a33*b33*k1*n1*n3*u3*v1 - 108*a33*b33*k1*n2**2*
u2*v2 - 216*a33*b33*k1*n2*n3*u2*v3 - 108*a33*b33*k1*n2*n3*u3*v2 - 216*a33*b33*k1
*n3**2*u3*v3 + 108*a33*b33*k10*n1**2*u3*v3 + 108*a33*b33*k10*n1*n3*u1*v3 + 108*
a33*b33*k10*n2*n3*u2*v3 + 108*a33*b33*k10*n3**2*u3*v3 - 162*a33*k1*n1**3*u1 -
162*a33*k1*n1**2*n2*u2 - 162*a33*k1*n1**2*n3*u3 + 162*a33*k1*n1*n3**2*u1 + 162*
a33*k1*n2*n3**2*u2 + 162*a33*k1*n3**3*u3 + 162*a33*k10*n1**3*u1 + 162*a33*k10*n1
**2*n2*u2 + 162*a33*k10*n1**2*n3*u3 - 162*a33*k10*n1*n3**2*u1 - 162*a33*k10*n2*
n3**2*u2 - 162*a33*k10*n3**3*u3 - b33**4*k1*v1**4 - 2*b33**4*k1*v1**2*v2**2 - 2*
b33**4*k1*v1**2*v3**2 - b33**4*k1*v2**4 - 2*b33**4*k1*v2**2*v3**2 + 6*b33**3*k1*
n1*v1*v3**2 + 6*b33**3*k1*n2*v2*v3**2 + 12*b33**3*k1*n3*v3**3 + 3*b33**3*k10*n1*
v1*v3**2 + 3*b33**3*k10*n2*v2*v3**2 - 9*b33**2*k1*n1**2*v2**2 + 18*b33**2*k1*n1*
n2*v1*v2 + 36*b33**2*k1*n1*n3*v1*v3 + 9*b33**2*k1*n2**2*v2**2 + 36*b33**2*k1*n2*
n3*v2*v3 - 36*b33**2*k1*n3**2*v1**2 - 36*b33**2*k1*n3**2*v2**2 + 18*b33**2*k10*
n1**2*v1**2 + 9*b33**2*k10*n1**2*v2**2 + 18*b33**2*k10*n1*n2*v1*v2 + 18*b33**2*
k10*n1*n3*v1*v3 + 9*b33**2*k10*n2**2*v2**2 + 18*b33**2*k10*n2*n3*v2*v3 + 27*b33*
k1*n1**3*v1 + 27*b33*k1*n1**2*n2*v2 + 54*b33*k1*n1**2*n3*v3 + 27*b33*k1*n1*n3**2
*v1 + 27*b33*k1*n2*n3**2*v2 - 27*b33*k10*n1**3*v1 - 27*b33*k10*n1**2*n2*v2 - 54*
b33*k10*n1**2*n3*v3 - 27*b33*k10*n1*n3**2*v1 - 27*b33*k10*n2*n3**2*v2)/(1296*a33
**4)$