Additive problems with the summands of a special type

The talk is devoted to the following additive problem. Suppose that $\alpha>1$ is a fixed irrational number. Let $r_3(\alpha,N)$ equals to the number of partitions of $N$ into a sum of two square -free summands and the term of the type $[\alpha q]$ with square -free $q$. In other words, $r_3(\alpha,N)$ is the number of representation $q_1+q_2+[\alpha q_3]=N$ where the numbers $q_1,q_2,q_3$ are square -free. Then the following asymptotic formula holds
$$ r_{3}(\alpha,N)\,=\,\frac{1}{2\alpha}\biggr(\frac{6}{\pi^2}\biggl)^{\!3}N^{2}+O\bigl(N^{11/6+\varepsilon}\bigr) $$
as $N\to\infty$.