Integer combinations of two irrational numbers equal to an integer

Let $x,y$ be two irrational numbers in $(0,1)$ with the property that for some non-zero integers $a,b,c$ we have $ax+by=c$. I have a general question and a more specific related question.

The general question is: what can we say about the set of all integers $c$ that can be written this way as an integer combination of $x$ and $y$? In particular must there be many such $c$ if we know that there is in particular one non-zero such $c$, or can it happen (for some $x,y$) that such a $c$ is unique?

The more specific question is: if there is one such combination $ax+by=c$, must there also exist a combination $a'x+b'y=c'$ where $a',b',c'$ are still non-zero integers and in addition $a',b'$ are coprime?