The question whether a differential equation admits nonconstant first integrals in the neighborhood of a stationary point is of considerable interest for its qualitative analysis. On the other hand, to detect the existence of a local first integral is frequently a difficult problem. In the present paper we discuss differential systems with analytic right-hand side, and approach the problem via Poincaré–Dulac normal forms: Thus we will obtain a clear picture of formal first integrals, and some nontrivial results about local analytic first integrals. Let $ U $ be an open neighborhood of $ 0 $ in $ \\mathbb{C} ^{n} $ and let $ f : U \\rightarrow \\mathbb{C} ^{n} $ be an analytic vector valued function with $ f (0) = 0 $ . In this work we discuss autonomous differential systems of the form $ $ \\dot{x}=f(x)=(f_{1}(x),\\cdots , f_{n}(x)) , \\quad x=(x_{1},\\cdots , x_{n}) $ $ Since real-analytic systems may be extended to $ \\mathbb{C} ^{n} $, our results will also be applicable to the real setting. To $ f $ there corresponds a vector field, i.e. a derivation of local analytic functions given by $$ \\chi _{f}(\\phi)=\\displaystyle\\sum_{i=0}^n{f_{i}}\\frac{\\partial\\phi}{\\partial x_{i}}=0.$$ We will employ differential equation and vector field interpretations simultaneously. To the commutator of two derivations there corresponds the Lie bracket of vector valued functions. If $ \\phi $ is a local analytic function then we call $ \\chi _{f}(\\phi) $ the Lie derivative of $\\phi $ . If $ \\chi _{f}(\\phi)=0 $ then we call $ \\phi $ a first integral of the differential equation. (Note that we include constant functions in this definition.)In this thesis we investiage dimansion of first integral's algebra and the form of formal (analytic) first integrals for local autonomous differential equation near a stationary point. We first decompose the jacobian matrix near fixed point, into semi simple and nilpotent part, i.e $ A=A _{s}+A _{n} $ and we find the first integrals of semi simple part. Now we can get number and form of local first integrals for system. One of the usefull implements for this purpose is the Poincare-Dulac normal forms. The existence question for formal first integrals , the ralation of normal forms and first integrals and convergence of normal forms by various examples for realization of matters will be discussed in that in chapter $ 3 $. We will proceed in chapter $ 4 $ to characterize and discuss the maximal scenario when all first integrals of As are conserved by the system transformed to normal form. Then we present a left; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" dir=ltr conserve all first integrals of the linear part and we discuss the case when some of the eigenvalues are zero in the linearization (with no other resonances), and we generalize known theorems relating local manifolds of stationary points to the existence of analytic first integrals . In chapter $ 5 $ b y $ sl(2) $ represention theory we express application of normal forms in first integral computing for Hopf, Takens -Bogdanov and Hopf-zero singularities.