  We present VLT/MUSE observations targeting the extended Lyman-$\alpha$
(Ly$\alpha$) emission of five high-redshift ($z\sim$3-4) submillimeter galaxies
(SMGs) with increasing quasar (QSO) radiation: two SMGs, two SMGs hosting a
QSO, and one SMG hosting a QSO with a SMG companion (QSO+SMG). These sources
should be located in dark matter halos of comparable masses (average mass of
$M_{\rm DM}\sim10^{12.2}\,{\rm M}_\odot$). We quantify the luminosity and
extent of the Ly$\alpha$ emission, together with its kinematics, and examine
four Ly$\alpha$ powering mechanisms: photoionization from QSOs/star formation,
shocks by galactic/QSO outflows, gravitational cooling radiation, and
Ly$\alpha$ photons resonant scattering. We find a variety of Ly$\alpha$
luminosities and extents, with the QSO+SMG system displaying the most extended
and bright nebula, followed by the SMGs hosting a QSO, and finally the
undetected circumgalactic medium (CGM) of SMGs. This diversity implies that
gravitational cooling is unlikely to be the main powering mechanism. We show
that photoionization from the QSO and QSO outflows can contribute to power the
emission for average densities $n_{\rm H}>0.5\,$cm$^{-3}$. Moreover, the
observed Ly$\alpha$ luminosities scale with the QSO's budget of Ly$\alpha$
photons modulo the dust content in each galaxy, highlighting a possible
contribution from resonant scattering of QSO's radiation in powering the
nebulae. We find larger Ly$\alpha$ linewidths (FWHM$\gtrsim
1200\,$km$\,$s$^{-1}$) than usually reported around radio-quiet systems,
pointing to large-scale outflows. A statistical survey targeting similar
high-redshift massive systems with known host properties is needed to confirm
our findings.
