Janvier and Arsenault 2007 took another look at
Euphanerops longaevus (Woodward 1900; Late Devonian, Figs. 1, 2) comparing it uncertainly to living lampreys and extinct jawless, finless fish. They report, “The anatomy of Euphanerops longaevus is reconstructed here on the basis of 17 specimens, 14 of which were hitherto undescribed. Practically all the mineralized elements that can be observed in the largest individuals of E. longevous display the same structure, which strikingly recalls that of lamprey cartilage, despite the uncertainty as to the origin of its mineralization.”
Elongated and confluent paired fins
“The new material of E. longaevus described here provides strong support for the presence of ventrolateral, ribbon-shaped, paired fins armed with numerous parallel radials. These fins extend from the anus to the anterior part of the branchial apparatus anteriorly, and are the first instance of paired fins with radials, whose anteroposterior extension largely overlaps that of the branchial apparatus in a vertebrate.”
Mostly true, but let’s not forget in manta rays and guitarfish, skates and rays, paired pectoral fins indeed do overlap the branchial apparatus (= gill basket), IF that is happening in Euphanerops (see below).
From the abstract
“Owing to the uncertainty as to the biogenic or diagenetic nature of the anatomical features described in E. longevous, no character analysis is proposed. Only a few possible homologies are uniquely shared by euphaneropids and either lampreys or anaspids, or both.”
Phylogenetically, the authors note:
“Euphanerops longaevus has been referred to as an anaspid, chiefly because of its distinctive hypocercal tail and anal fin. However, since it apparently has no mineralized dermal skeleton, E. longaevus lacks evidence for the tri-radiate postbranchial spine, which Forey (1984) proposed as the defining character of the Anaspida. Consequently, it is now often treated in recent phylogenetic analyses as a separate terminal taxon, alongside other scale-less (or “naked”) jawless vertebrate taxa also once regarded as anaspids, namely Endeiolepis and Jamoytius.”

Figure 1. Several basal chordates: Branchiostoma, Euphanerops, Jamoytius and Birkenia. The middle image of Euphanerops is the tracing. The others are freehand interpretations from Janvier and Arsenault 2007.
Here
(Fig. 2) individual skull bones and tiny overlooked pectoral and pelvic fins are identified. Adding a missing (unossified?) rostrum (= nasal) restores the original profile. In the large reptile tree (LRT, 1717+ taxa) Euphanerops nests basal to sturgeons, like Pseudoscaphirhynchus (FIg. 3), a clade not mentioned by Janvier and Arsenault 2007. A previously enigmatic element in front of the mouth is here identified as a pair of barbels, as in sturgeons. The tiny dorsal spines of Euphanerops are also found as larger dorsal armor in Birkenia, osteostracans and sturgeons.

Figure 7. Transitional Euphanerops is more sturgeon-like, but the location of the operculum and/or gill openings are not apparent here. The nasal is missing from both the plate and counter plate.
According to Wikipedia
Euphaneropidae have, “greatly elongated branchial apparatus which covers most of the length of the body.”
Here that area is identified as a typical subdivided and flattened ventral surface, as in Birkenia, sturgeons and osteostracans.

Figure 2. Skull of Pseudoscaphorhynchus. Note the mouth is created by the lacrimal and angular, not the maxilla and dentary, which are tooth-bearing bones in more derived fish.
The hypocercal tail of Euphanerops
has heterocercal elements and this taxon nests between taxa with a heterocercal tail. With an Ordovician genesis, Late Devonian Euphanerops likely developed a dipping tail and larger propulsive dorsal fin secondarily, as a reversal. An ancestor, Birkenia, has a similar dipping tail.

Figure 4. Euphanerops caudal fin with elements re-identified.
Small enigmatic squares of rod-like elements near the cloaca
are here identified as primitive pelvic fins or vestiges of the same. More primitive taxa do not have pelvic fins. More derived taxa do.

Figure 5. Euphanerops with elements here identified as tiny pectoral fins just anterior to the cloaca and posterior to the ventral armor. Images from Janvier and Arsenault 2007.
Primitive pectoral fins
are known in ancestral and descendant taxa, so Euphanerops should have them, too. Here (Fig. 6) they are identified as vestiges.

Figure 6. Euphanerops plate and counter plate with colors added identifying elements.
Traditionally sturgeons have not been tested with osteostracans
(Fig. 7) and other jawless fish. The LRT tests a wide gamut of competing candidates and nests sturgeons prior to the advent of jaws and teeth in vertebrates, close to osteostracans and Euphanerops. Do not let one or two traits, like a dipping (hypocercal) tail, steer you off course in your wide-gamut analysis.

Figure 7. Top to bottom: Thelodus a soft jawless fish with a ventral oral opening and gill slits, perhaps a hint of diamond-shaped armor laterally. Hemicyclaspis, adds extensive armor. Euphaneropsps, a late survivor of an Ordovician radiation basal to sturgeons. Acipenser, a sturgeon with a protrusible tube for a mouth and reduced armor. Chondrosteus, a fish with jaws, but no marginal teeth.
The ‘paired fin ridges’ observed by Janvier and Arsenault
may be ray-like ossifications that gathered to produce the ventrolateral armor on sturgeons (Fig. 7) or were vestiges thereof. Additionally, that’s where basal chordate gonads are located.
A set of lamprey-like gill openings appear near the skull
of Euphanerops. This appears to be a retention of or reversal back to similar multiple openings seen in Birkenia (Fig. 1). Again, don’t judge a taxon by one or two traits. Test them all against a wide gamut of taxa, like the LRT. We may be seeing what happens a the transition from multiple gill openings to a sturgeon-like operculum here.
References
Janvier P, Desbiens S, Willett JA and Arsenault 2006. Lamprey-like gills in a gnathostome related Devonian jawless vertebrate. Nature 440:1183–1185.
Janvier P and Arsenault M 2007. The anatomy of Euphanerops longaevus Woodward, 1900, an anaspid-like jawless vertebrate from the Upper Devonian of Miguasha, Quebec, Canada. Geodiversitas 29 (1) : 143-216.
Woodward AS 1900. On a new ostracoderm fish (Euphanerops longaevus) from the Upper Devonian of Scaumenac Bay, Quebec, Canada. Magazine of Natural History ser. 7, 5: 416-419.
wiki/Euphaneropidae