classical mechanics - Does the conservation of $frac{partial L}{partialdot{q}_i}$ necessarily require $q_i$ to be cyclic?

If a generalized coordinate $q_i$ is cyclic, the conjugate momentum $p_i=\frac{\partial L}{\partial\dot{q}_i}$ is conserved.

Is the converse also true? To state more explicitly, if a conjugate momentum

$$p_i=\frac{\partial L}{\partial\dot{q}_i}=C_1\tag{1}$$ is conserved, will $q_i$ be necessarily cyclic? If we integrate $(1)$, we get $$L=C_1(q_i,\dot{q}_i)\dot{q}_i+C_2(q_i) q_i\tag{2}$$ From $(2)$, it is evident that the conservation of $p_i$ does not necessarily imply $q_i$ is cyclic. $q_i$ is cyclic only if $C_2=0$ which is only a special case.

Assuming my little observation is correct what is an example (perhaps a physical one) of such a situation i.e., a conserved $p_i$ with a non-cyclic $q_i$? I cannot immediately think of one.