of motion \[3.17\] F\^\_[tx]{}=\_t A\^\_x-\_x A\^\_t+\[A\^\_t,A\^\_x\]= \_t A\^\_x-\_x A\^\_t= \_t A\^\_x-{H(u,v),A\^\_x}=0 Given a curve $c$ on $M=\Rl\times S^1$ we define its holonomy as the usual path ordered product \[3.18\] h\_c(A\^)=[P]{}(\_c dx A\^) where $\cal P$ denotes path ordering with the lowest parameter value to the outmost left. Given a parameterization of the path $[0,b]\to c;\;s\mapsto c(s)$ the holonomy is the unique solution to the ordinary differential equation \[3.18a\] h\_[c\_s]{}(A\^)=h\_[c\_s]{}(A\^) A\^\_(c(s)) \^(s), h\_[c\_0]{}(A\^):=1\_N,h\_[c\_b]{}(A\^) h\_c(A\^) where $c_s=c([0,s])$.

Supposing that the curvature of $A$ vanishes, that is, that $A$ is flat, it is clear that for loops $c$ on $\Rl\times S^1$ only those which are not contractible lead to a non – trivial value of $h_c(A)$. Among the non contractible loops of winding number one all are homotopic to the loop $c_{t,x}=\{t\}\times [x,x+2\pi)$ in the worldsheet time slice $t=$const. winding once around the cylinder with starting point $x$. In the theory of integrable systems, the holonomy of that loop is called the monodromy matrix \[3.19\] h\_[t,x]{}(A\^):=h\_[c\_[t,x]{}]{}(A\^) Notice that (\[3.19\]) is not a periodic function of $x$ even if the connection $A$ is flat.

Consider an arbitrary interior point $x_0$ of $c_{t,x}$. This point subdivides the loop into two edges $e_{t,x}=\{t\}\times [x_0,x]$ and $e'_{t,x}=\{t\}\times 