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-Szöke, Szöke 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 first author's work "On maximal totally real embeddings" [Pali]. In our series of articles we focus on the case of torsion free connections. In the present article we provide a fiber-wise 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 general settings that can be useful for the study of open questions in the theory of the embeddings in consideration. In this article we also provide evidence for some remarkable canonical vanishing of some of the integrability equations in quite general settings.