Let $E \rightarrow M$ be a plane bundle endowed with an almost complex structure $J.$ $J$ induces a natural positive definite inner product in the associated bundle $$End(E)\rightarrow M,$$ denoted by $<,>.$ Now consider the set $$P=\{(p,A)| \,\ p \in M, \,\ A \in End(E_p), \,\ <A,J>=0, \,\ <A,id>=0, <A,A>=1 \}$$
$P$ is a circle fibration above $M.$ Are there any conditions on $J$ that will guarantee $P$ is trivial?