Airport Time Capsule not seen by Airport Utility

Based on your observations and comments, I would venture to say that your TC was damaged by an electrical surge during that lightning storm ... even if your other equipment faired better.

If you haven't already, you can try performing a "factory default" reset on the TC as your last option before replacing it.

Ref:  How to reset your AirPort base station - Apple Support

As Bob has mentioned, your Arris device is a combination modem & wireless router. These are also known as a gateway device. Regardless, your TC is also a wireless router, but does not have a built-in modem. Folks that typically have both of these devices, use the TC primarily for Time Machine backups. However, to be properly configured, the TC would need to be reconfigured as a bridge and connected to the Arris directly by Ethernet. All of your current devices should be able to connect to this gateway, either by Ethernet or by wireless.

In this case, it appears that your TC has failed. There is no need to further troubleshoot it. If you do still need a destination for Time Machine backups, you have a few choices to consider:

• Replace the TC with another one. Understand that Apple stopped selling these about 2 years ago. You may still find them by searching the Internet, but I wouldn't really recommend doing so as these are outdated devices.
• Connect an external drive to your Mac.
• Use a Network Storage Device (NAS) that supports Time Machine backups. Once source for these would be Synology.

my home network ions.

I think you meant icons.. but ions could be the cause of the problem.. When hit by the electrons in a big lighting bolt then everything inside the TC is turned to ions.. very bad.. ions also smell.. like burned wires. Have a sniff up close through the ethernet ports or the bottom vents. A distinct burnt smell and well its all over red rover.

Test it by plugging ethernet on the TC to your router or computer that you know is working.. Test each port of the TC in turn.. the yellow connectivity light should turn on the side of the port. Does not matter for now which port.. try all 4.. connectivity will work whether LAN or WAN. Sometimes a port that was not connected can survive.

The fact that the light blinks slowly (?? rapid blinking means it is dead) shows the damage is perhaps restricted to the ethernet ports.. you can possibly rescue it for something.. likely it can no longer be used as a router. Tell us what you find.

Did you have to replace the main modem??

Any suggestions on how together my router back to working?

Replacement!!

Any suggestions on how together my router back to working?

First things first. If there was a nearby lightning strike, the voltage surge would have probably taken out your modem first before it traveled to the Time Capsule.

So, your first check would be to unplug the Time Capsule from your modem and then connect your Mac or other computer directly to the modem to see if you can get a good Internet connection that way.

If you have not already done this, please perform this check and post back on your results, since that will dictate the next step(s) with the Time Capsule.

Would also help if you could provide the make and model number of your modem, since there may be an alternate way to set up the Time Capsule depending on the type of modem that you have.

My TC continues to have rapid amber flashing signal.

Not a good sign at all. That should be a slow blinking amber light about once a second.

Unplug the TC from power and let it rest for awhile.....a hour or two.

Then, try a Factory Default Reset

If the TC is operating correctly, it will display a slow blinking amber light about once a second.

Post back on your progress.

You already have a router.......the Arris TG2472......which is technically a modem/router, also called a gateway. This is a device that combines a separate modem and a separate router in one single package.

You don't need another router unless you are saying that the Arris TG2472 is also not operating correctly.

We're also drifting off topic, which is your original question about the Time Capsule. Do you want to try to continue to troubleshoot that device, or are you convinced that the TC has been damaged?

Two previous posters have suggested performing a Factory Default Reset again before you declare the Time Capsule dead. Do you want to do this, or are you giving up on the Time Capsule?

You don't need another router. Connect WiFi devices to the network that your Arris gateway is providing. If you want to make backups of your Mac, simply connect a USB hard drive to the Mac and back up that way.

If you want to stay with an Apple AirPort product, there are new, refurbished and used Time Capsules available on eBay.com.