  Background: In highly deformed nuclei, there is a noticeable coupling of the
Isoscalar Giant Monopole Resonance (ISGMR) and the $K = 0$ component of the
Isoscalar Giant Quadrupole Resonance (ISGQR), which results in a double peak
structure of the isoscalar monopole (IS0) strength (a narrow low-energy
deformation-induced peak and a main broad ISGMR part). The energy of the narrow
low-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 the
ISGMR in the prolate $^{24}$Mg and oblate $^{28}$Si nuclei and identify among a
variety of energy density functionals based on Skyrme parameterisations the one
which best describes the experimental data. This will allow for conclusions
regarding the nuclear incompressibility. Because of the strong IS0/IS2
coupling, 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 detect
and momentum analyse the inelastically scattered $\alpha$ particles. An
experimental energy resolution of $\approx 70$ keV (FWHM) was attained,
revealing fine structure in the excitation-energy region of the ISGMR. The IS0
strength distributions in the nuclei studied were obtained with the
Difference-of-Spectrum (DoS) technique. The theoretical comparison is based on
the quasiparticle random-phase approximation (QRPA) with a representative set
of Skyrme forces.
