In the LRT
Desmostylus (Fig 1) nests at the base of the baleen whales, the clade Mysticeti.
Figure 1. Desmostylus in several views. Hypothetical baleen added in red. Rostrum distortion straightened. Note the ear hole = external auditory meatus (dark yellow) rising and incorporating into the squamosal (magenta). Note the width of the mandible compared to the hourglass rostrum as in mysticetes shown in figue 2. In palatal view the baleen is spread laterally. In reality the baleen would appear as a series of points, not lines.
Unfortunately
academic whale experts universally agree that mysticetes arose from odontocetes, but that only works by excluding outgroup taxa tested in the LRT.
Here
hypothetical baleen is added to Desmostylus (Fig 1) descending from a toothless micro-ridge inset from the lateral edge of the palate. This is where premaxillary teeth used to be. Note the mandible is wider than the rostrum in Desmostylus as in Caperea and Eubalaena, (Fig 2) two examples of baleen whales.
After testing all competing candidates, this is where and how baleen developed in desmostylians prior to the transformation of fore limbs, the loss of hind limbs and development of flukes.
Figure 2. Taxa in the lineage of right whales include Desmostylus, Caperea and Eubalaena. The tiny bit of jugal posterior to the orbit (in cyan) is found in all baleen whales tested so far. The frontals over the eyes are just roofing the eyeballs in Desmostylus, much wider in Caperea and much, much longer in Eubalaena.
Desmostylus hesperus
(Marsh 1888; Oligocene. Figs 1, 2) was the first desmostylian discovered and is known from several distinct species. One has a relatively larger skull and smaller limbs with a smaller hind limb and thus is transitional to right whales like Caperea and Eubalaena. That specimen also has smaller anterior teeth and narrower jaw tips. Like all mysticetes, the mandible is wider than the rostrum. A complete skeleton is shown to scale with sister taxa (Fig 2).
References
Cope ED 1872. Descriptions of some new Vertebrata from the Bridger Group of the Eocene. Proceedings of the American Philosophical Society 12:460-465.
Domning DP, Ray, CE and McKenna, MC 1986. Two new Oligocene desmostylians and a discussion of Tethytherian systematics. Smithsonian Contributions to Paleobiology. 59. pp. 1–56.
Fordyce RE and Marx FG 2013. The pygmy right whale Caperea marginata: the last of the cetotheres. Proceedings of the Royal Society B: Biological Sciences 280(1753):1–6.
Marsh OC 1888. Notice of a new fossil sirenian, from California. American Journal of Science 25(8):94–96.
Peters D 2018. The triple origin of whales rejected manuscript. Online here.
Reinhart RH 1959. A review of the Sirenia and Desmostylia. University of California Publications in Geological Sciences 36(1):1–146.
Santos G, Parham J and Beatty B 2016. New data on the ontogeny and senescence of Desmostylus (Desmostylia, Mammalia). Journal of Vertebrate Paleontology. doi: 10.1080/02724634.2016.1078344
Tsai C-Hi and Fordyce RE 2015. Ancestor–descendant relationships in evolution: origin of the extant pygmy right whale, Caperea marginata. Biol Lett. 2015 Jan; 11(1): 20140875.
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