  We study the autonomous systems of quadratic differential equations of the
form $\dot{x}_i(t)=\mathbf{x}(t)^T \mathbf{A}_i \mathbf{x}(t) + \mathbf{v}_i^T
\mathbf{x}(t)$ with $\mathbf{x}(t) = (x_1(t),x_2(t),\dots,x_i(t),\dots)$ which,
in general, cannot be solved exactly. In the present paper, we present a
subclass of analytically solvable quadratic systems, whose solution is realized
through a multi-dimensional generalization of the inversion which transforms a
quadratic system into a linear system. We provide a constructive algorithm
which, on one hand, decides whether the system of differential equations is
analytically solvable with the inversion transformation and, on the other hand,
provides the solution. The presented results apply for arbitrary, finite number
of variables.
