Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero for both $J_N$ and $J_M$).

Does it imply that there can not exist any non constant pseudo-holomorphic map $~f:N\rightarrow M$? Pseudo-holomorphic means $$ df\circ J_M = J_N\circ df. $$

If this is true, can someone point out a reference for this fact? I am aware that vanishing of the Nijenhuis tensor is a necessary and sufficient condition for an integrable ACS, but I don't see immediately why its non vanishing implies the non existence of non constant pseudo-holomorphic maps.

Secondly, what if $J_N$ is integrable, but $J_M$ is not? (In my original question, both $J_N$ and $J_M$ were non integrable.)

Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero for both $J_N$ and $J_M$).

Does it imply that there can not exist any non constant pseudo-holomorphic map $~f:N\rightarrow M$? Pseudo-holomorphic means $$ df\circ J_M = J_N\circ df. $$

If this is true, can someone point out a reference for this fact? I am aware that vanishing of the Nijenhuis tensor is a necessary and sufficient condition for an integrable ACS, but I don't see immediately why its non vanishing implies the non existence of non constant pseudo-holomorphic maps.

Secondly, what if $J_N$ is integrable, but $J_M$ is not and the real dimension of $N$ is greater than two?

Note that in my original question, both $J_N$ and $J_M$ were non integrable. Secondly, non constant pseudo-holomorphic curves exist from $(\Sigma,j) $ to $(M, J_M)$ when $(\Sigma,j)$ is an almost complex manifold of real dimension two (i.e. a Riemann Surface). Hence, I added the condition that real dimension of $N$ is greater than two.