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The reader is referred to [EG] for more details. Let (M, V) be a smooth involutive structure with ﬁber dimension of V over C equal to n. 4. A smooth submanifold X of M is called maximally real if CTp M = Vp ⊕ CTp X for each p ∈ X. Note that if (M, V) is CR, then X is maximally real if and only if it is totally real of maximal dimension. If X is a maximally real submanifold and p ∈ X, deﬁne VpX = {L ∈ Vp : L ∈ Tp X}. 5. 1 in [EG]) V X is a real subbundle of V|X of rank n. The map : VX → T M which takes the imaginary part induces an isomorphism VX ∼ = T M|X /T X.

Consider the C 1 mapping Ψ(x) = (x, u(x), ux (x)) that maps U into (x, ζ0 , ζ) space. It is easy to see that the push forwards Ψ∗ (Luj ) agree with HF◦j on the class of C 1 functions that are holomorphic in (ζ0 , ζ). The lemma follows. 2. The holomorphic Poisson bracket satisﬁes the Jacobi identity. That is, {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0. Proof. Observe ﬁrst that the sum {f, {g, h}} + {g, {h, f }} + {h, {f, g}} is a ﬁrstorder diﬀerential operator in each function since for example it equals Hf ({g, h})+ [Hg , Hh ](f ).

HFj , HFk ] = l=1 Proof. 2, [HFj , HFk ] = H{Fj ,Fk } and by hypothesis, {Fj , Fk } = 0 on Σ. The lemma follows from these equations. 4. 3. Let V u be the bundle generated by Lu1 , . . , Lun . Then V u is involutive. Proof. 3 implies for the principal parts HF◦j that n [HF◦i , HF◦j ] alij HF◦l = l=1 For each i, j = 1, . . , n, write m [Lui , Luj] = crij (x) r=1 ∂ . 1) 40 S. 1)) l=1 n = alij (x, u(x), ux (x))Lul (xr ). l=1 linear combination of the Luk . 5. 1. Then near each point p ∈ E, we can get real analytic coordinates (x, t), x = (x1 , .