Anisotropic flows in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Au</mml:mi> <mml:mo>+</mml:mo> <mml:mi>Au</mml:mi> </mml:mrow> </mml:math> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:

Within the framework of the lattice Boltzmann-Uehling-Uhlenbeck transport model, we present a systematic study of proton anisotropic flow observables measured by the HADES collaboration, by utilizing the recently developed density-, momentum- and isospin-dependent N$5$LO Skyrme pseudopotential. In particular, we investigate the impacts of the momentum dependence of nucleon mean-field potentials, the stiffness of symmetric nuclear matter (SNM) EOS, the high-density behaviors of the symmetry energy and the in-medium modification of nucleon-nucleon elastic cross sections on proton $v_{1}$, $v_{2}$, $v_{3}$, and $v_{4}$ in Au+Au collisions at $\sqrt{s_{\rm{NN}}} = 2.4\,\text{GeV}$. Our results show that the proton anisotropic flows are strongly sensitive to the momentum dependence of nucleon mean-field potential as well as the incompressibility coefficient $K_0$ of SNM. In addition, the transverse momentum dependence of the proton $v_2$ exhibits a modest sensitivity to the higher-order skewness coefficient $J_0$ and kurtosis coefficient $I_0$ of SNM as well as the momentum dependence of the symmetry potential, while the transverse momentum dependence of the proton $v_1$ is shown to modestly depend on the in-medium modification of nucleon-nucleon elastic cross sections. Moreover, the high-density symmetry energy seems to have limited effects on the proton anisotropic flows. These findings highlight the necessity of considering the momentum dependence of nucleon mean-field potentials including the symmetry potential, the higher-order characteristic parameters of SNM EOS beyond $K_0$, and the in-medium modification of nucleon-nucleon elastic cross sections, in future Bayesian transport model analyses on proton anisotropic flows in heavy-ion collisions at HADES energies, thereby to extract information on nuclear matter EOS as well as the associated underlying nuclear effective interactions.