On the Number of Integer Points Whose First Coordinates Satisfy a Divisibility Condition on Hyperboloids of a Special Form

The discrete ergodic method is applied to obtain an asymptotic expression for the number of all integer points in a given bounded domain on a three-dimensional hyperboloid of genus determined by the invariants $[w,2]$, where $w$ is odd, such that the first coordinates of these points are divisible by $w$.