Propositional dynamic logic (PDL) is an important modal logic used to specify and reason about the behavior of software. A challenging problem in the context of PDL is solving fixed-point equations, i.e., formulae of the form x≡φ(x) such that x is a propositional variable and φ(x) is a formula containing x. A solution to such an equation is a formula ψ that omits x and satisfies ψ≡φ(ψ), where φ(ψ) is obtained by replacing all occurrences of x with ψ in φ(x). In this paper, we identify a novel class of PDL formulae arranged in two dual hierarchies for which every fixed-point equation x≡φ(x) has a solution. Moreover, we not only prove the existence of solutions for all such equations, but also provide an explicit solution ψ for each fixed-point equation.