In Introduction to Automata Theory, Languages, and Computation, John Hopcroft mentioned
In fact, a PDA In fact the languages of one counter machines are accepted by deterministic PDA's although the proof is surprisingly complex.
How to prove that?
In contrast, is there a language that is accepted by some DPDA but can't be accepted by any non-deterministic one counter machines?
: John E. Hopcroft and Rajeev Motwani and Jeffrey D. Ullman (2006). Introduction to Automata Theory, Languages, and Computation. Boston/MA: Addison Wesley. P359.