= Solution
Suppose a $C^1$ <classical solution> existed and set $w=u+iv$. The system is precisely
$$
u_x=v_t,
\qquad
u_t=-v_x,
$$
which are the <Cauchy-Riemann equations> for $w$ as a function of the <complex number> $z=x+it$. Hence $w$ is <holomorphic function>[holomorphic] near $z=0$. Every holomorphic function is <real analytic function>[real analytic], so its restriction
$$
w(x)=w(0,x)=f(x)+ig(x)
$$
to the real axis is real analytic near zero. Its <real part> and <imaginary part> show that both $f$ and $g$ must be real analytic, contradicting the hypothesis. Therefore no such $C^1$ solution exists.
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