  This article is the continuation of the first named author work "On maximal
totally real embeddings". For real analytic compact manifolds equipped with a
covariant derivative operator acting on the real analytic sections of its
tangent bundle, a construction of canonical maximal totally real embeddings is
known from previous works by Guillemin-Stenzel, Lempert, Lempert-Sz{\"o}ke,
Sz{\"o}ke and Bielawski. The construction is based on the use of Jacobi fields,
which are far from being explicit. As a consequence, the form of the
corresponding complex structure has been a mystery since the very beginning. A
quite simple recursive expression for such complex structures has been provided
in the above cited first named author work. In our series of papers we always
focus on the torsion free case. In the present paper we provide a fiberwise
Taylor expansion of the canonical complex structure which is expressed in terms
of symmetrization of curvature monomials and a rather simple and explicit
expression of the coefficients of the expansion. Our main argument applies to
far more generals settings that can be useful for the study of open questions
in the theory of the embeddings in consideration. In this paper we provide also
evidence for some remarkable canonical vanishing of some of the integrability
equations in general settings.
