= Solution
The <coordinate basis> vector $K=\partial_v$ has constant components $K^a=\delta^a_v$. Its <norm> is
$$
g_{ab}K^aK^b=g_{vv}=0,
$$
so it is a nonzero <null vector> everywhere. The <Lie derivative> of the <metric tensor> is
$$
(\mathcal L_Kg)_{ab}
=K^c\partial_cg_{ab}+g_{cb}\partial_aK^c+g_{ac}\partial_bK^c
=\partial_vg_{ab}=0.
$$
Thus $\boxed{K=\partial_v\text{ is a null Killing vector field}.}$ In fact it is a <parallel vector field>: $K_b=g_{bv}=\delta_b^u$ and the <Christoffel symbols> found below give $\nabla_aK_b=-\Gamma^u{}_{ab}=0$. This stronger property is characteristic of this <plane-fronted gravitational wave>.
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