Find all possible a and b for which α is nonempty

Question

Let $a$ and $b$ be two integers. The relation $\alpha\subseteq \mathbb{Z}\times\mathbb{Z}$ is defined by $x \alpha y$ if and only if $аx+by=1$.

Find all possible $a$ and $b$ for which $α$ is nonempty.

For all $а$ and $b$ proof existence or absence of each of following properties: irreflexive, symmetric, antysymmetric and transitive.

Here is my attempt [link]