function defines an automorphism of the Poisson algebra of smooth functions on phase space, we obtain for any smooth function of the functional form $F=F[Y_+,Y_-]$ that \[3.8\] \[\^+\_u(t)\](F)=F\[(\^u\_t)\^Y\_+,Y\_-\], \[\^-\_u(t)\](F)=F\[Y\_+,(\^u\_t)\^Y\_-\] Writing $\alpha^\pm_{\varphi^u_t}:=\alpha^\pm_u(t)$ we may extend (\[3.8\]) to all elements $\varphi$ of Diff$(S^1)$ \[3.9\] \[\^+\_\](F)=F\[\^Y\_+,Y\_-\], \[\^-\_\](F)=F\[Y\_+,\^Y\_-\] Besides this local gauge freedom of the string we also have global Poincaré symmetry. The generators of infinitesimal translations and Lorentz transformations respectively are given by \[3.10\] p\_&=& \_[S\^1]{} dx \_\
J\^ &=& \_[S\^1]{} dx \[X\^\^ \_-X\^\^ \_\] It is straightforward to check that $p_\mu,J^{\mu\nu}$ have vanishing Poisson brackets with the $V_\pm(u)$ and thus are strong Dirac observables. Moreover, we have \[3.11\] {p\_,Y\^\_(x)}=0, {J\^,Y\^\_(x)}=( \^ Y\^\_-\^ Y\^\_)(x) Hence for the corresponding Hamiltonian flows we obtain \[3.12\] \_a(Y\_)&:=&\_[n=0]{}\^ {a\^p\_,Y\_}\_[(n)]{}=Y\_\
\_(Y\_)&:=&\_[n=0]{}\^ {\_ J\^,Y\_}\_[(n)]{}= (\_ \^) Y\_ where $\Lambda$ is an antisymmetric matrix and $\tau^{\mu\nu}$ are appropriate basis elements of $so(1,D-1)$. Denoting $L=\exp(\Lambda_{\mu\nu} \tau^{\mu\nu})$ the automorphisms extend to smooth functions of $Y_\pm$ as \[3.13\] \_a(F)=F,\_L(F)=F\[LY\_+,LY\_-\]

Algebra of Invariants {#s3.2}
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This finishes our analysis of the actions of the constraints and symmetry generators on the $Y_\pm$. We will now construct Dirac observables $F[Y_+,Y_-]$ by using the theory of integrable systems. The idea is to use the method of Lax 