Today President Obama designated the Organ Mountains-Desert Peaks region near Las Cruces, NM to be a National Monument. The 500,000 acre area includes mountain, desert and grassland habitats as well has archeological and historical resources. There are numerous petroglyph sites and the Butterfield Stage route passed through the area. Several rare plant and animal species also call the area home. There are hiking, biking and horse trails but currently I was unable to find any information about regulations for pets.
Here is a link to the website: Organ Mountains. When I get some photos and some information about pet regulations I will update this post. If any of my readers are in that area and can find out if there are any pet friendly trails please post the information in the comment section.