In a space \(C^ n\) an integrable system of differential Pfaff equations of the form \(df=\omega f\) is considered, where f is an unknown p- dimensional vector of complex-valued functions, \(\omega =\sum^{m}_{i=1}A_ i\frac{dL_ i}{L_ i}\) is a matrix differential form, \(A_ i\) are constant commutative complex \(p\times p\)-matrices. It is supposed that on the reducible complex manifold \(L=\cup^{m}_{i=1}L_ i\), the components \(L_ i\) of which are non- singular and have only normal intersections, the system has regular singularities. The existence of the exponential representation of the fundamental matrix of the system in the form \(\phi (x)=e^{\Omega (x)}\) is proved, where matrix \(\Omega\) (x) is expanded in a series, whose elements are from an infinite system of differential equations. Let on an algebraic manifold a function f(x) of Nilsson class with manifold branching \(L\subset X\) be given. It is proved that in the neighbourhood of simple points and in the neighbourhood of points of normal intersection of the components of the manifold L, the vector \(\theta f=(f,Df,...,D_{p-1}f)\) satisfies a certain system of differential Pfaff equations with regular singularities.