  We study the integrable system of first order differential equations
$\omega_i(v)'=\alpha_i\,\prod_{j\neq i}\omega_j(v)$, $(1\!\leq i, j\leq\! N)$
as an initial value problem, with real coefficients $\alpha_i$ and initial
conditions $\omega_i(0)$. 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 ${\rm d}v^*=g(\omega_i)\,{\rm d}v$ that separate
geometry from dynamic. Some of those parametrizations turn out to be
generalization of the {\sl 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.
