Counting the number of solutions to an equation involving floor and ceiling function

Question

I'd like to count the number of integer solutions to $x \lceil \mu y \rceil - y\lfloor \mu x \rfloor=K$, where $\lceil \cdot \rceil$ is the ceiling function, $\lfloor \cdot \rfloor$ is the floor function, $\mu$ is a given irrational number and $K$ is a given positive integer. Especially, I'm interested in those "good" positive solutions $(x,y)$ satisfying that $(x-y,y)$ is NOT a solution to the equation. I've tried some small values of $\mu$ and $K$, and found that the number of "good" positive solutions may be $\sigma_1(K)$, where $\sigma_1(\cdot)$ is the sum of positive divisor function. Any insight about the problem is appreciated.