# Question:CLOSED FORM SOLUTION OF ISOSPECTRAL FLOW EQUATIONS

## Question:CLOSED FORM SOLUTION OF ISOSPECTRAL FLOW EQUATIONS

Maple

To anyone out there who may be able to help

I have been researching into closed-form solutions of the 2 x 2 case of the Matrix Differential Equations

dM(s)/ds = [ M(s) , D(s)/2 ]   ,    M(0) = Mo in sl(2,R)  ( 1 )

dD(s)/ds = [ M(s) , K(s) ]      ,     D(0) = Do in sl(2,R)  ( 2 )

dK(s)/ds = [ D(s)/2 , K(s) ]    ,    K(0) =  Ko in sl(2,R)  ( 3 )

where s is a real-valued parameter.

Now if

M(s) := [ [ m[1](s) , m[2](s) ] , [ m[3](s) , -m[1](s) ] ]  ( 4 )

D(s) := [ [ d[1](s) , d[2](s) ] , [ d[3](s) , -d[1](s) ] ]        ( 5 )

K(s) := [ [ k[1](s) , k[2](s) ] , [ k[3](s) , -k[1](s) ] ]        ( 6 )

dm[1](s)/ds = ( m[2](s).d[3](s) - m[3](s)*d[2](s) )/2  ,  m[1](0) = m1o  ( 7 )

dm[2](s)/ds = m[1](s).d[2](s) - m[2](s).d[1](s)           ,  m[2](0) = m2o  ( 8 )

dm[3](s)/ds = m[3](s).d[1](s) - m[1](s).d[3](s)            , m[3](0) = m3o ( 9 )

dd[1](s)/ds =  m[2](s).k[3](s) - m[3](s)*k[2](s)            , d[1](0) = d1o ( 10 )

dd[2](s)/ds = 2.(m[1](s).k[2](s) - m[2](s).k[1](s) )       , d[2](0) = d2o ( 11 )

dd[3](s)/ds = 2.(m[3](s).k[1](s) - m[1](s).k[3](s) )       , d[3](0) = d3o ( 12 )

dk[1](s)/ds = ( d[2](s).k[3](s) - d[3](s)*k[2](s) )/2        , k[1](0) = k1o ( 13 )

dk[2](s)/ds = d[1](s).k[2](s) - d[2](s).k[1](s)                  , k[2](0)  = k2o ( 14 )

dk[3](s)/ds = d[3](s).k[1](s) - d[1](s).k[3](s)                  , k[3](0)  = k3o ( 15 )

which constiutes a sytem of nine coupled quadratic homogeneous ordinary differential equations.

It is readily shown that this system has the following first integrals

m[1](s)^2 + m[2](s).m[3](s) == m1o^2 + m2o.m3o  ( 16 )

2.m[1](s).d[1](s) + m[2](s).d[3](s) + m[3](s).d[2](s)  = 2.m1o.d1o + m2o.d3o + m3o.d2o  ( 17 )

2.m[1](s).k[1](s) + m[2](s).k[3](s) + m[3](s).k[2](s) + d[1](s)^2 + d[2](s).d[3](s)

2.m1o.k1o + m2o.k3o + m3o.k2o + d1o^2 + d2o.d3o  ( 18 )

2.d[1](s).k[1](s) + d[2](s).k[3](s) + d[3](s).k[2](s)  = 2.d1o.k1o + d2o.k3o + d3o.k2o  ( 19 )

k[1](s)^2 + k[2](s).k[3](s) == k1o^2 + k2o.k3o  ( 20)

My first question is are there any other linearly independent first integrals and can Maple ( I have Maple 13 ) be used to find them ? ? ?

Both Maple 13 and Mathematica 7 fail to solve the nine coupled quadratic homogeneous ordinary differential equations.