Invariant subspaces in some function spaces on symmetric spaces. III

We describe the structure of closed linear subspaces in some topological vector spaces that consist of functions on the exceptional non-compact symmetric space $M=F_4/{\operatorname{Spin}(9)}$ (the Cayley space) and are invariant under the natural quasi-regular representation of the group $F_4$. The class of function spaces under consideration contains the spaces $C^d(M)$ of $d$-times continuously differentiable functions ($d=0,1,\dots,\infty$) and the spaces of functions of exponential growth on $M$.