Maurer-Cartan equation in the DGLA of graded derivations

Let M be a smooth manifold and D = L-Psi + I-Psi a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Psi, Psi are differential 1-form on M with values in the tangent bundle TM and L-Psi, I-Psi are the d* and i* components of D. Under the hypothesis that Id(T(M)) + Psi is invertible we prove that Psi = b (Psi) = - 1/(2) under bar (Id(TM) + Psi)(-1) omicron [Psi, Psi](FN), where [., .](FN) is the Frolicher-Nijenhuis bracket. This yields to a classification of the canonical solutions e(Psi) = L-Psi + I-b(Psi) of the Maurer-Cartan equation according to their type: e(Psi) is of finite type r if there exists r is an element of N such that Psi(r) omicron [Psi, Psi](FN) = 0 and r is minimal with this property, where [., .](FN) is the Frolicher-Nijenhuis bracket. A distribution xi subset of TM of codimension k >= 1 is integrable if and only if the canonical solution e(Psi) associated to the endomorphism Psi of TM which is trivial on xi and equal to the identity on a complement of xi in TM is of finite type <= 1, respectively of finite type 0 if k = 1.