What rule allows Amateur Operators to establish and operate remote, automated stations like APRS weather stations or beacons?
Are there different rules specifically for beacons, or do they all fall under the same rules?
There's a few parts:
§97.109 Station control. (d) When a station is being automatically controlled, the control operator need not be at the control point. Only stations specifically designated elsewhere in this part may be automatically controlled. Automatic control must cease upon notification by a District Director that the station is transmitting improperly or causing harmful interference to other stations. Automatic control must not be resumed without prior approval of the District Director.
§97.113 Prohibited transmissions. (d) No amateur station, except an auxiliary, repeater, or space station, may automatically retransmit the radio signals of other amateur station.
§97.213 Telecommand of an amateur station.
An amateur station on or within 50 km of the Earth's surface may be under telecommand where:
(a) There is a radio or wireline control link between the control point and the station sufficient for the control operator to perform his/her duties. If radio, the control link must use an auxiliary station. A control link using a fiber optic cable or another telecommunication service is considered wireline.
(b) Provisions are incorporated to limit transmission by the station to a period of no more than 3 minutes in the event of malfunction in the control link.
(c) The station is protected against making, willfully or negligently, unauthorized transmissions.
(d) A photocopy of the station license and a label with the name, address, and telephone number of the station licensee and at least one designated control operator is posted in a conspicuous place at the station location.
Beacons have their own rules, which are too numerous to just copy/paste. The section is entitled §97.203 Beacon station., and can be found at ARRL.