I will present my question in the context that I encountered it, although I believe it probably applies in general context. Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form $\omega = \sum dp_k\wedge dq_k$. Consider a Lagrangian $L\cong S^1 \times S^1$ given by the product of two circles, $A$ and $B$, each lying in an $\mathbb{R}^2$ factor of $\mathbb{R}^4$. Consider a class $\alpha\in\pi_2(\mathbb{R}^4,L)$, such that $\omega(\alpha)>0$.

1) Does there exists an almost complex structure $J$ compatible with $\omega_{std}$ such that $\alpha$ is represented by a $J$-holomorphic disc $u$ with boundary mapped to $L$?

Note that for the integrable complex structure $J_{std}$ that comes from $\mathbb C$ this is true only for $\alpha$ of the form $kA+mB$, where $k,m\ge0$ (here I abused notation and referred to the discs bounded by the circles $A$ and $B$ using the same letters)

2) If the answer to 1 is positive, can $J$ be chosen to be regular? (in the sense that all the moduli spaces of curves are being cut out transversally).

Thank you! :)