Integrability of an almost complex structure vs holomorphity of the $ M rightarrow mathcal {J} (M) $ section

Let's say we have an almost complex variety. $ (M, J) $. Consider the complex vector package $ V rightarrow M $ whose fiber over $ x $ It is the space of almost complex structures on $ T_x M $.

Is there any logical connection between the following two conditions:

  • $ J $ it is integrable;
  • the map $ M rightarrow V $ associated to $ J $ Is it holomorphic?