Neyman–Pearson lemma for non monotonic spaces

Question

Question: Does the Neyman–Pearson lemma give instructions for how to construct the test when the outcome space is not monotonic?

I suspect the answer is NO, but I would like to:

1. Get an affirmative answer that it is indeed NO.
2. If you have an alternative lemma for such cases (instead of going through all of the possible rejection regions), I'd be curious to know about it.

An extended example for illustration:

Assume an observation from a multinomial distribution with options {a,b,c}, with the following two null hypothesis about the probabilities:

$H_0:p=(1/3, 1/6, 1/2)$

$H_1:p=(2/3, 2/6, 0)$

The likelihood ratio for each of the possible outcomes is:

$\lambda(a) = 2$

$\lambda(b) = 2$

$\lambda(c) = 0$

Which means we can build rejection regions using one of the following rules:

$\lambda > 3 => never\ reject => \alpha=0$

$\lambda > 1 => reject\ for\ a,b => \alpha=1/2$

$\lambda >= 0 => always\ reject => \alpha=1$

So I know that if I am interested in the most powerful test for $\alpha=1/2$, I know what the test is. But what if I want a test for $\alpha=1/3$? I can use either "reject if a", or "reject if b", and obviously the first one is more powerful ($\pi = 2/3$), but I can think of this rule by looking at my options, not through the NP lemma.