LookAt on a single relative axis

Hello,
I need some help with a big problem which i can’t resolve :

I have a turret with the pattern that make it following the player on a single axis. It work well with this code :

Vector3 target_position = this.target.transform.position;
target_position.y = this.transform.position.y;
this.transform.LookAt(target_position);

However, i want this pattern working with local rotation of the turret, in that event the axis for the rotation can be another axis. So, how can i get a local constraint for the rotation ?

Thank you for you help,

Julien J.

Anyone please ?