pairs, that is, one reformulates the equations of motion for $Y_\pm$ as a matrix equation of the form $M_{,\alpha}=[A_\alpha,M]$ where $M(t,x)$ is an $N\times N$ matrix and $A_\alpha$ is a “connection”, that is, a one form on $\Rl\times S^1$ with values in $GL(N,\Cl)$. The integrability condition for the Lax pair $L,A$ is the zero curvature equation $F_{\alpha\beta}=
\partial_\alpha A_\beta
-\partial_\beta A_\alpha+[A_\alpha,A_\beta]=0$ and one looks for $L,A$ in such a way that $F_{\alpha\beta}=0$ is equivalent to the equations of motion. Given such a setup, it follows that the functions Tr$(L^n),\;
n=1,..,N$ are constants of the motion ($n>N$ leads to algebraically dependent invariants due to the theorem of Hamilton – Caley).

We proceed to the details. The “Hamiltonian” of the string is given by \[3.14\] H(u,v)=(C(u)+D(v))= V\_+(v+u)+V\_-(v-u) where $u,v$ are arbitrary test functions. “Time” evolution is defined by \[3.15\] \_:={H(u,v),Y\_}=\[(vu) Y\_\]’ Notice that the “time” evolution depends on $u,v$ and it will be our task to show that our final invariants constructed by the Lax pair method do not depend on $u,v$.

Let $\tau_I$ be a basis of $GL(n,\Rl)$ and $T_\mu^I$ some complex valued matrices. We define the following connection \[3.16\] A\^\_x:=Y\^\_T\_\^I \_I=:Y\^\_T\_, A\^\_t:=(vu) A\^\_x The zero curvature condition reproduces the equations 