= Solution
The scalar <Cauchy-Kovalevskaya theorem> says that a <partial differential equation> solved for its highest derivative normal to a <real analytic function>[real-analytic] <non-characteristic hypersurface>, with real-analytic coefficients and <Cauchy data>, has a unique local real-analytic solution. In coordinates, an equation
$$
\partial_t^m\phi=F\bigl(t,x,(\partial_t^j\partial_x^\alpha\phi)_{j<m}\bigr)
$$
has such a solution near the origin when $F$ and the prescribed values of $\partial_t^j\phi(0,x)$ for $0\leq j<m$ are real analytic.
Choose a real-analytic primitive $F_0$ of $f$ near zero and apply the theorem to the scalar <Laplace equation>
$$
\phi_{tt}=-\phi_{xx},
\qquad
\phi(0,x)=F_0(x),
\qquad
\phi_t(0,x)=-g(x).
$$
The line $t=0$ is <non-characteristic hypersurface>[non-characteristic] because the coefficient of $\phi_{tt}$ is one. Define
$$
u=\phi_x,
\qquad
v=-\phi_t.
$$
Then $u(0,x)=f(x)$ and $v(0,x)=g(x)$, while equality of mixed derivatives and the Laplace equation give
$$
u_t=\phi_{xt}=-v_x,
\qquad
v_t=-\phi_{tt}=\phi_{xx}=u_x.
$$
Thus $(u,v)$ is the required local real-analytic solution.
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