Need help adding fan to existing 3-way switch setup

Question

I'm including a diagram, because it is difficult to describe all of the things going on here. I have 2 sets of 3-way switches on either side of a room. 1 set operates some can lights with a switch on one side of the room and a dimmer on the other side. The other set has a switch on either side of the room and looks like it was once wired up to a fan. I'm attempting to add a new fan, but i'm not sure how to wire it up with the existing setup. The new fan has no light and comes with a wall switch that just has the hot, neutral, and ground. It looks like this:

From what I can tell the is a line coming into the middle of the box, then each 3-way is wired up from there up to the ceiling. In the ceiling where the new fan should be, I have 2 of the 3-wire (white,black, and red) cables. Here is my diagram of the switches on the hot side (I've excluded the grounds for the sake of simplicity):

So, question #1 is: Can I use this fan switch with the existing setup?

And question #2: How would I go about this?

Answer 1

If the fan is controlled by a 3-way switch, you'll have to replace either of the switches with another 3-way switch. Unless you eliminate one of the 3-way switches, you can't install a single pole switch.

If you want to install a fan speed control switch in this scenario, you'll have to find one that can be wired as a 3-way switch (not sure if it exists).