= Solution
Use the mostly-plus <Minkowski metric>, and write $\approx$ for equality on the <constraint surface>. Assume that the <mechanical constraints> are locally independent. They are <first-class constraints> when
$$
\{\varphi_i,\varphi_j\}=C_{ij}{}^k(q,p)\varphi_k.
$$
Thus their <Poisson brackets> vanish on the <constraint surface>, and their <Hamiltonian> flows preserve that surface. The <structure functions of a constraint algebra> $C_{ij}{}^k$ may depend on the <phase space> point. \b[The finite real span of the constraints is a <Lie algebra> if it closes with constant structure coefficients], in a suitable choice of generators. The <Jacobi identity> then gives the usual conditions on the <structure constants of a Lie algebra>. With general structure functions the finite real span need not close, even though the <Poisson bracket> of all smooth functions is itself a <Lie bracket>.
To see the <gauge invariance> directly, let $G=\epsilon^i(t)\varphi_i$ generate a <canonical gauge transformation>:
$$
\delta q^I=\{q^I,G\},\qquad
\delta p_I=\{p_I,G\},\qquad
\boxed{\delta\lambda^k=\dot\epsilon^k-\lambda^i\epsilon^jC_{ij}{}^k.}
$$
The variation of the <phase-space action> integrand is
$$
\delta(p_I\dot q^I-\lambda^i\varphi_i)
=\frac{d}{dt}(p_I\delta q^I-G)
+\left(\dot\epsilon^k-\delta\lambda^k-\lambda^i\epsilon^jC_{ij}{}^k\right)\varphi_k.
$$
The second term cancels without using the <equations of motion>. Taking $\epsilon^i$ to vanish at the temporal boundaries leaves the <action> invariant. Arbitrary functions $\epsilon^i(t)$ therefore relate different descriptions of the same physical motion. This reasoning also works with structure functions; constant structure coefficients are only needed for the finite-dimensional <Lie algebra> claim.
For a <closed string>, choose $0\leq\sigma<2\pi$ and periodic fields. A convenient <Nambu-Goto phase-space action> is
$$
I=\int dt\,d\sigma\left[P_m\dot X^m-\frac e2(P^2+T^2X'^2)-vP\cdot X'\right].
$$
Here $e$ and $v$ are <Lagrange multipliers>. The <Nambu–Goto phase-space constraints> are $\mathcal H_0=(P^2+T^2X'^2)/2\approx0$ and $\mathcal H_1=P\cdot X'\approx0$, with canonical <Poisson brackets>
$$
\{X^m(\sigma),P_n(\sigma')\}=\delta^m_n\delta_{2\pi}(\sigma-\sigma').
$$
Let $J_\pm^m=P^m\pm TX'^m$. Differentiating the periodic <Dirac delta function> gives
$$
\{P^m(\sigma),X'^n(\sigma')\}
=\eta^{mn}\partial_\sigma\delta_{2\pi}(\sigma-\sigma'),
\qquad
\{X'^m(\sigma),P^n(\sigma')\}
=\eta^{mn}\partial_\sigma\delta_{2\pi}(\sigma-\sigma').
$$
The opposite signs in $J_+$ and $J_-$ cancel these terms, so \b[$\{J_+^m(\sigma),J_-^n(\sigma')\}=0$]. Replace the original <constraints> by the equivalent chiral densities
$$
\mathcal H_\pm=\frac{J_\pm^2}{4T},\qquad
\mathcal H_0=T(\mathcal H_++\mathcal H_-),\qquad
\mathcal H_1=\mathcal H_+-\mathcal H_-.
$$
Their mixed <Poisson brackets> vanish. Choose opposite Fourier orientations for the two sectors:
$$
L_n=\int_0^{2\pi}d\sigma\,e^{in\sigma}\mathcal H_+(\sigma),\qquad
\widetilde L_n=\int_0^{2\pi}d\sigma\,e^{-in\sigma}\mathcal H_-(\sigma).
$$
The <chiral constraint algebra of a closed string> is
$$
\boxed{\{L_m,L_n\}=-i(m-n)L_{m+n},\quad
\{\widetilde L_m,\widetilde L_n\}=-i(m-n)\widetilde L_{m+n},\quad
\{L_m,\widetilde L_n\}=0.}
$$
Each is the <Witt algebra>: the <vector fields> $\ell_n=ie^{in\sigma}\partial_\sigma$ on a circle satisfy $[\ell_m,\ell_n]=(m-n)\ell_{m+n}$. Fourier expansion identifies each real algebra, with $L_n^*=L_{-n}$, with the <Lie algebra> of <vector fields> on the circle. The two commuting copies give \b[$\mathrm{Diff}_1\oplus\mathrm{Diff}_1$], not a quantum central extension.
For an <open string>, allowed <boundary conditions> must remove the endpoint term in the variation of the <action>, consistently with the allowed endpoint variations. The spatial boundary term is
$$
\delta I\big|_{\partial\sigma}=-\int dt\,
\left[(eT^2X'_m+vP_m)\delta X^m\right]_{\sigma=a}^{\sigma=b}.
$$
It expresses the <open-string endpoint momentum flux>. In the <temporal gauge for a string> $X^0=t$, take a boundary-adapted parametrization with $v=0$ at the ends. Fixing $X^i(t,a)=0$ gives $\delta X^i(t,a)=0$, a <Dirichlet boundary condition>. At the other end allow arbitrary spatial variations; for nonzero $e$ these require $X'^i(t,b)=0$, a <Neumann boundary condition>. Also $X'^0=0$ in this <temporal gauge for a string>, so $X'^m(t,b)=0$. The <constraint> at this <free-end string boundary condition> reduces to $P^2=0$. Hamilton's equation $\dot X=eP+vX'$ consequently gives $\dot X^2=0$ there. Since $\dot X^0=1$, \b[the free endpoint has spatial speed one]. This is the <null motion of a free string endpoint>.
A <straight rotating string with one fixed endpoint> supplies the required solution in at least two spatial dimensions. Set $e=1/T$, $v=0$, and
$$
\boxed{X^0=t,\qquad
\boldsymbol X(t,\sigma)=L\sin(\sigma/L)
\big(\cos(t/L),\sin(t/L),0,\ldots\big),\quad
0\leq\sigma\leq\frac{\pi L}{2}.}
$$
Take $P_m=T\dot X_m$. The <Hamilton's equations> become $\ddot X=X''$, which holds because both second derivatives give $-\boldsymbol X/L^2$. The <Nambu–Goto phase-space constraints> are satisfied by
$$
\dot X\cdot X'=0,\qquad
\dot X^2+X'^2=-1+\sin^2(\sigma/L)+\cos^2(\sigma/L)=0.
$$
The endpoint at $\sigma=0$ stays at the origin, while $X'=0$ at $\sigma=\pi L/2$ and the endpoint moves around a circle of radius $L$ with <angular speed> $1/L$. At each time the whole string lies on a straight radial segment. Its spatial <proper length> is
$$
\int_0^{\pi L/2}|\boldsymbol X'|\,d\sigma
=\int_0^{\pi L/2}\cos(\sigma/L)\,d\sigma=L.
$$
The <velocity> is everywhere perpendicular to the segment, so this also equals the sum of local rest-frame lengths. The induced <worldsheet metric> becomes degenerate at the null free endpoint, as expected for the limiting free-end solution.
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