two nondegeneracy conditions. Moreover, the techniques involved in all these works have also been used in analyzing many other dispersive equations or systems which are not necessarily Klein-Gordon, see for example [@GM11], [@GMS09], [@GMS12], [@GMS4], [@IP11], [@IP13].

In this paper we will completely solve the small data problem of (\[nlkg\]) in $3D$, removing the assumptions made in [@IP12]. More precisely, we will prove the following

\[mainkg\] Consider the system (\[nlkg\]) in $\mathbb{R}\times\mathbb{R}^3$. We make the following assumptions.

1.  The nonlinearity $\mathcal{Q}_{\alpha}$ in (\[nlkg\]) has the form$$\label{symmetrykg}\mathcal{Q}_{\alpha}(u,\partial u,\partial^{2}u)=\sum_{\beta,\gamma=1}^{d}\sum_{j,k,l=1}^{3}\big(A_{\alpha\beta\gamma}^{jk}u_{\gamma}+B_{\alpha\beta\gamma}^{jkl}\partial_{l}u_{\gamma}\big)\partial_{j}\partial_{k}u_{\beta}+\mathcal{Q}_{\alpha}'(u,\partial u),$$ where $A$ and $B$ are tensors symmetric in $\alpha$ and $\beta$ with norm not exceeding one, and $\mathcal{Q}_{\alpha}'$ is an arbitrary quadratic form (of constant coefficient) of $u$ and $\partial u$;

2.  The initial data $u(0)=g,\partial_{t}u(0)=h$ satisfy the bound$$\label{initialkg}\|(g,\partial_{x}g,h)\|_{\mathcal{H}^{N}}+\|(g,\partial_{x}g,h)\|_{Z}\leq \varepsilon\leq\varepsilon_{0},$$ where $N=1000$, the norm $$\label{normkg}\|u\|_{\mathcal{H}^{k}}=\sup_{|\beta|\leq k}\|\Gamma^{\beta}u\|_{L^{2}},\qquad\Gamma=(\partial_{i},x_{i}\partial_{j}-x_{j}\partial_{i})_{1\leq i<j\leq 3},$$ the $Z$ norm is defined in Definition \[normdef\], and $\varepsilon_0$ is small enough depending on $b_{\alpha}$ and $c_{\alpha}$.

Then we have the followings:

1.  The system (\[nlkg\]) has a unique solution $u$, with prescribed initial data, such that $$u\in C_{t}^{1}\mathcal{H}_{x}^{N}(\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{d}),\quad\partial_{x}u\in C_{t}^{0}\mathcal{H}_{x}^{N}(\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{d});$$

2.  We have energy bound and decay estimates $$\label{bound01}\|u(t)\|_{\mathcal{H}^{N}}+\|(\partial_{x},\partial_{t})u(t)\|_{\mathcal{H}^{N}}\lesssim\varepsilon,$$ $$\label{bound02}\sum_{|\mu|\leq N/2}\big(\|\Gamma^{\mu}u(t)\|_{L^{\infty}}+\|\Gamma^{\mu}(\partial_{x},\partial_{t})u(t)\|_{L^{\infty}}\big)\lesssim\frac{\varepsilon}{(1+|t|)\log^{20}(2+|t|)};$$

3.  There exist $\mathbb{R}^{d}$ valued functions $w^{\pm}$ verifying the linear equation $$(\partial_{t}^{2}-c_{\alpha}^{2}\Delta+b_{\alpha}^{2})w_{\alpha}^{\pm}=0,$$ 