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We study the integrable system of first order differential equations ω i (v) = α i j =i ω j (v), (1 ≤ i, j ≤ N ) as an initial value problem, with real coefficients α i and initial conditions ω i (0). The analysis is based on its quadratic first integrals. For each dimension N , the system defines a family of functions, generically hyperelliptic functions. When N = 3, this system generalizes the classic Euler system for the reduced flow of the free rigid body problem, thus we call it N -extended Euler system (N -EES). In this Part I the cases N = 4 and N = 5 are studied, generalizing Jacobi elliptic functions which are defined as a 3-EES. Taking into account the nested structure of the N -EES, we propose reparametrizations of the type dv * = g(ω i ) dv that separate geometry from dynamic. Some of those parametrizations turn out to be generalization of the Jacobi amplitude. In Part II we consider geometric properties of the N -system and the numeric computation of the functions involved. It will be published elsewhere.keywords: Integrable systems Generalized Euler system Jacobi and Weierstrass elliptic functions third Legendre elliptic integral arXiv:1505.06142v1 [math.DS]