Check a variable within a range with a binary variable
If I understand corectly, you want to formulate a system of linear constraints on a real variable $a$ and a $\{0,1\}$-variable $y$ 
such that the solutions are given by
$$[0,1]\times\{1\} \bigcup [1,\infty)\times \{0\}.$$

This is impossible.  No linear constraint can put an upper bound on $a$ for some values of $y$ without putting an upper bound on $a$ for all values of $y$.

You can, however, just split the problem into two pieces and solve each one separately.