basis $e_1(x), e_2(x)$ of $L_x$ such that $\omega_x(
e_1(x), e_2(x)) > 0$, and an orthonormal basis $e_3(x), e_4(x)$ of $N_x$ such that the basis $e_1(x), e_2(x), e_3(x), e_4(x)$ gives the same orientation of $T_xV_1$ as $\omega^2$.

Define an almost complex structure $J^1$ on $U_1$ by $J^1e_1 = e_2$, $J^1 e_3 =
e_4$. Then $J^1$ depends smoothly on $x$, even when $e_j(x)$ are not smooth in $x$. Furthermore, we have $\omega _x(e_1(x), J_x^1e_1(x)) > 0$ and thus from [*Lemma 1.1.1*]{} we see that the relation $\omega _x(e_3(x), J_x^1e_3(x)) > 0$ is also satisfied. This means that our $J^1$ is tamed by $\omega $. Note also that $M$ is $J^1$-holomorphic.

Denote by $\jj_x$ the sphere of $g$-orthogonal complex structures on $T_x
X$ as in [*Lemma 1.4.1*]{}. Let $\gamma_x$ be the shortest geodesic on $\jj_x$ joining $J(x)$ - our given integrable structure, with $ J_x^1$. Put $J_t(x)
= J^1_{\gamma _x(t\cdot \Vert \gamma _x\Vert \cdot \phi (x ))}$. Here $\Vert
\gamma_x \Vert $ denotes the length of $\gamma _x$, $\phi$ is smooth on $X$ with support in $U_1$ and identically one in the neighborhood of $M$. The curve $\{ J_t \}$ satisfies all the conditions of our lemma.

Lemma 1.4.3. [*Under the conditions of 