%Toda lattice and Lax pair;
%Program name: toda.our;
linelength 72;
off nat$
out todal$

matrix L(3,3), A(3,3), Lt(3,3);

% Lax pair;
L := mat((b1, a1, 0), (a1, b2, a2), (0, a2, b3));
A := mat((0, -a1, 0), (a1, 0, -a2), (0, a2, 0));

% equation of motion;
Lt := A*L - L*A;

b1t := Lt(1,1);
b2t := Lt(2,2);
b3t := Lt(3,3);
a1t := Lt(1,2);
a2t := Lt(2,3);

I1 := trace(L);     % first integral;
I2 := trace(L*L);   % first integral;

% ansatz;
I := b1*b2*b3-b1*a2*a2-b3*a1*a1;
% check that I is a first integral;
res := b1t*df(I,b1) + b2t*df(I,b2) + b3t*df(I,b3)
       + a1t*df(I,a1) + a2t*df(I,a2);

%can this first integral be found with tr(L^n) ?;

write ";end"$
shut todal$
on nat$