at $p$ satisfy a linear system of ordinary differential equations. Since $W(p, 0) = 0$, it follows that $(\nabla J)(p, \tau) = W(p, \tau) = 0$ for all $\tau$.

But, if $J$ is parallel and $g$ is Hermitian relative to $J$, we then have ${\operatorname{Rc}}\circ J = J\circ {\operatorname{Rc}}$ as endomorphisms of $TM$. So $${{\frac{\partial }{\partial\tau}}}{\operatorname{Rc}}= J\circ {\operatorname{Rc}}- {\operatorname{Rc}}\circ J = 0,$$ and $J(\cdot, \tau) = J(\cdot, 0)$ on $M\times [0, T]$.

More generally, if $g(\tau)$ is a solution to the backward Ricci flow for which the reduced holonomy remains fixed (e.g., if $(M, g(\tau))$ is complete and of bounded curvature) then any tensor $V_0$ which is parallel on $M$ with respect to $g(0)$ can be extended to a smooth family $V(\tau)$ of $g(\tau)$-parallel tensors on $M\times[0, T]$ via $D_{\tau} V = 0$ and $V(0) = V_0$.

Kählerity near spatial infinity
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Now we are ready to prove the first assertion in Theorem \[thm:kaehlerity\]. Our strategy is fundamentally the same as in [@KotschwarWangConical]: from the asymptotically conical soliton, we construct a self-similar solution to the backward Ricci flow defined for $\tau\in (0, 1]$ on a sufficiently distant end, which (after adjustment by a suitable diffeomorphism) converges smoothly as $\tau\searrow 0$ 