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Background: In highly deformed nuclei, there is a noticeable coupling of theIsoscalar Giant Monopole Resonance (ISGMR) and the $K = 0$ component of theIsoscalar Giant Quadrupole Resonance (ISGQR), which results in a double peakstructure of the isoscalar monopole (IS0) strength (a narrow low-energydeformation-induced peak and a main broad ISGMR part). The energy of the narrowlow-lying IS0 peak is sensitive to both the incompressibility modulus$K_\infty$ and the coupling between IS0 and isoscalar quadrupole (IS2)strength.  Objective: This study aims to investigate the two-peaked structure of theISGMR in the prolate $^{24}$Mg and oblate $^{28}$Si nuclei and identify among avariety of energy density functionals based on Skyrme parameterisations the onewhich best describes the experimental data. This will allow for conclusionsregarding the nuclear incompressibility. Because of the strong IS0/IS2coupling, the deformation splitting of the ISGQR will also be analysed.  Methods: The ISGMR was excited in $^{24}$Mg and $^{28}$Si using$\alpha$-particle inelastic scattering measurements acquired with an $E_\alpha= 196$ MeV beam at scattering angles $\theta_{\text{Lab}} = 0^\circ$ and$4^\circ$. The K$600$ magnetic spectrometer at iThemba LABS was used to detectand momentum analyse the inelastically scattered $\alpha$ particles. Anexperimental energy resolution of $\approx 70$ keV (FWHM) was attained,revealing fine structure in the excitation-energy region of the ISGMR. The IS0strength distributions in the nuclei studied were obtained with theDifference-of-Spectrum (DoS) technique. The theoretical comparison is based onthe quasiparticle random-phase approximation (QRPA) with a representative setof Skyrme forces.

Isoscalar giant monopole resonance in $^{24}$Mg and $^{28}$Si: Effect of coupling between the isoscalar monopole and quadrupole strength