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We present VLT/MUSE observations targeting the extended Lyman-$\alpha$(Ly$\alpha$) emission of five high-redshift ($z\sim$3-4) submillimeter galaxies(SMGs) with increasing quasar (QSO) radiation: two SMGs, two SMGs hosting aQSO, and one SMG hosting a QSO with a SMG companion (QSO+SMG). These sourcesshould be located in dark matter halos of comparable masses (average mass of$M_{\rm DM}\sim10^{12.2}\,{\rm M}_\odot$). We quantify the luminosity andextent of the Ly$\alpha$ emission, together with its kinematics, and examinefour Ly$\alpha$ powering mechanisms: photoionization from QSOs/star formation,shocks by galactic/QSO outflows, gravitational cooling radiation, andLy$\alpha$ photons resonant scattering. We find a variety of Ly$\alpha$luminosities and extents, with the QSO+SMG system displaying the most extendedand bright nebula, followed by the SMGs hosting a QSO, and finally theundetected circumgalactic medium (CGM) of SMGs. This diversity implies thatgravitational cooling is unlikely to be the main powering mechanism. We showthat photoionization from the QSO and QSO outflows can contribute to power theemission for average densities $n_{\rm H}>0.5\,$cm$^{-3}$. Moreover, theobserved Ly$\alpha$ luminosities scale with the QSO's budget of Ly$\alpha$photons modulo the dust content in each galaxy, highlighting a possiblecontribution from resonant scattering of QSO's radiation in powering thenebulae. We find larger Ly$\alpha$ linewidths (FWHM$\gtrsim1200\,$km$\,$s$^{-1}$) than usually reported around radio-quiet systems,pointing to large-scale outflows. A statistical survey targeting similarhigh-redshift massive systems with known host properties is needed to confirmour findings.

Circumgalactic Ly$\alpha$ emission around submillimeter-bright galaxies with different quasar contributions