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We study the Hamiltonian dynamics of the spherical spin model withfully-connected two-body interactions drawn from a Gaussian probabilitydistribution. In the statistical physics framework, the potential energy is ofthe so-called $p=2$ spherical disordered kind. Most importantly for oursetting, the energy conserving dynamics are equivalent to the ones of theNeumann integrable system. We take initial conditions in thermal equilibriumand we subsequently evolve the configurations with Newton dynamics dictated bya different Hamiltonian. We identify three dynamical phases depending on theparameters that characterise the initial state and the final Hamiltonian. Weobtain the {\it global} dynamical observables with numerical and analyticmethods and we show that, in most cases, they are out of thermal equilibrium.We note, however, that for shallow quenches from the condensed phase thedynamics are close to (though not at) thermal equilibrium. Surprisingly enough,for a particular relation between parameters the global observables complyGibbs-Boltzmann equilibrium. We next set the analysis of the system with finitenumber of degrees of freedom in terms of $N$ non-linearly coupled modes. Weevaluate the mode temperatures and we relate them to the frequency-dependenteffective temperature measured with the fluctuation-dissipation relation in thefrequency domain, similarly to what was recently proposed for quantumintegrable cases. Finally, we analyse the $N-1$ integrals of motion and we usethem to show that the system is out of equilibrium in all phases, even forparameters that show an apparent Gibbs-Boltzmann behaviour of globalobservables. We elaborate on the role played by these constants of motion inthe post-quench dynamics and we briefly discuss the possible description of theasymptotic dynamics in terms of a Generalised Gibbs Ensemble.

Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model