Kleeneliness is next to Gödeliness
A language $L$ 1 cannot be deterministically recognized by empty stack 2 if there exist two distinct strings of $L$ such that one is a prefix of the other.
In more formal terms: if $\exists u, v \in L, u \neq v$, such that $v$ is prefixed by $u$. The concept of prefix can be further formalized as $v = ux$, where $x$ is a string $(\neq \epsilon)$ of the alphabet to which both strings $u$ and $v$ belong.
The usefulness of this property is obvious as it allows us to realize that if we are trying to develop an elementary parser using a PDA for a given language that contains strings of the type aabb and aabbbb, the automaton will stop, by empty stack, as soon as the prefix is consumed.