I am certain that the following result holds, but was not able to find a reference. Do you know one?

Statement. Let $(M^4,\omega)$ be a compact symlectic manifold and let $J_t$ be a smooth family of compatible almost complex structures, $t\in(-1,1)$. Suppose that $S_0$ is a smooth $J_0$-holomorphic sphere in $(M^4,J_0)$ with zero self-intersection.

Consider now $M^4\times (-1,1)$, introduce $J_t$ on each fibre $M^4\times t$, and take in it the surface $S^2_0\times 0$. Then there is a neighbourhood $U$ of $S^2_0\times 0$ such that for any $(x,t)\in U$ there is a unique $J_t$-holomorphic sphere passing through $(x,t)$ contained in $U$. In other words $J_t$-holomorphic spheres produce on $U$ a structure of a smooth $S^2$-fibration (over a $3$-ball).

I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof?

Statement. Let $(M^4,\omega)$ be a compact symlectic manifold and let $J_t$ be a smooth family of compatible almost complex structures, $t\in(-1,1)$. Suppose that $S_0$ is a smooth $J_0$-holomorphic sphere in $(M^4,J_0)$ with zero self-intersection.

Consider now $M^4\times (-1,1)$, introduce $J_t$ on each fibre $M^4\times t$, and take in it the surface $S^2_0\times 0$. Then there is a neighbourhood $U$ of $S^2_0\times 0$ such that for any $(x,t)\in U$ there is a unique $J_t$-holomorphic sphere passing through $(x,t)$ contained in $U$. In other words $J_t$-holomorphic spheres produce on $U$ a structure of a smooth $S^2$-fibration (over a $3$-ball).

I would be grateful for a reference to a result from which the following statement follows.

Statement. Let $(M^4,\omega)$ be a compact symlectic manifold and let $J_t$ be a smooth family of compatible almost complex structures, $t\in(-1,1)$. Suppose that $S_0$ is a smooth $J_0$-holomorphic sphere in $(M^4,J_0)$ with zero self-intersection.

Consider now $M^4\times (-1,1)$, introduce $J_t$ on each fibre $M^4\times t$, and take in it the surface $S^2_0\times 0$. Then there is a neighbourhood $U$ of $S^2_0\times 0$ such that for any $(x,t)\in U$ there is a unique $J_t$-holomorphic sphere passing through $(x,t)$ contained in $U$. In other words $J_t$-holomorphic spheres produce on $U$ a structure of a smooth $S^2$-fibration (over a $3$-ball).

I am certain that the following result holds, but was not able to find a reference. Do you know one?

Statement. Let $(M^4,\omega)$ be a compact symlectic manifold and let $J_t$ be a smooth family of compatible almost complex structures, $t\in(-1,1)$. Suppose that $S_0$ is a smooth $J_0$-holomorphic sphere in $(M^4,J_0)$ with zero self-intersection.

Consider now $M^4\times (-1,1)$, introduce $J_t$ on each fibre $M^4\times t$, and take in it the surface $S^2_0\times 0$. Then there is a neighbourhood $U$ of $S^2_0\times 0$ such that for any $(x,t)\in U$ there is a unique $J_t$-holomorphic sphere passing through $(x,t)$ contained in $U$. In other words $J_t$-holomorphic spheres produce on $U$ a structure of a smooth $S^2$-fibration (over a $3$-ball).