  We calculate the mean total number of equilibrium points in a system of $N$
random autonomous ODE's introduced by Cugliandolo et al. in 1997 to describe
non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we
suggest a new approach which allows such a calculation to be done most
straightforwardly, and is based on efficiently incorporating the Langrange
multiplier into the Kac-Rice framework. Analysing the asymptotic behaviour for
large $N$ we confirm that the phenomenon of 'topology trivialization' revealed
earlier for other systems holds also in the present framework with
nonrelaxational dynamics. Namely, by increasing the variance of the random
'magnetic field' term in dynamical equations we find a 'phase transition' from
the exponentially abundant number of equilibria down to just two equilibria.
Classifying the equilibria in the nontrivial phase by stability remains an open
problem.
