Raising and Lowering Operators of a Hamiltonian

Lets say that I have a Hermitian Hamiltonian $H$ with a non-Hermitian raising operator operator $A$ which satisfies
begin{equation}
[H,A] = Omega A, quad Omega in mathbb{R}_{>0}.
end{equation}

Then it is fairly straightforward to show that $H$ has a symmetry in the form
begin{equation}
[H, AA^{dagger}] = 0.
end{equation}

My question is can I use these statements, and only these statements, to say anything about the relationship between the operators $A$ and $AA^{dagger}$? Ideally I would like to be able to show that if I have some eigenstate $vert a rangle$ where $AA^{dagger}vert a rangle = a vert a rangle$ then $(AA^{dagger})Avert a rangle = (a+c_{a})Avert a rangle$ with $c_{a}$ some scalar number dependent on the $vert a rangle$ at hand, i.e. $A$ acts as a raising/ lowering operator to move between eigenstates of $AA^{dagger}$.

Obviously there are a number of systems where this is true as these are relationships typically satisfied by ladder operators which appear frequently in Quantum Mechanics. But I am unsure if the first two equations are enough to make statements about the relationship between $A$ and $AA^{dagger}$.