  We generalize and refine the hyperk\"ahler Arnold conjecture, which was
originally established, in the non-degenerate case, for three-dimensional time
by Hohloch, Noetzel and Salamon by means of hyperk\"ahler Floer theory. In
particular, we prove the conjecture in the case where the time manifold is a
multidimensional torus and also establish the degenerate version of the
conjecture. Our method relies on Morse theory for generating functions and a
finite-dimensional reduction along the lines of the Conley-Zehnder proof of the
Arnold conjecture for the torus.
