On equivariant topological complexity of smooth $\zmod{p}$-spheres

We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear $\zmod{p}$-spheres have both invariants either $2$ or $3$ and calculate exact values in all but two cases for linear actions. On the other hand, we exhibit examples which show that these invariants can be arbitrarily high in the class of smooth $\zmod{p}$-spheres.