Perseus analyses of ἁλίοιο:

ᾱλίοιο,ἅλιος 1 (of the sea): masc/neut gen sg (epic)
ᾱλίοιο,ἅλιος 1 (of the sea): masc/fem/neut gen sg (epic)
ἅλιος 2 (fruitless, idle): masc/neut gen sg (epic)
ἅλλομαι (spring, leap): fut opt mid 2nd sg (doric)
ᾱλίοιο,ἥλιος (sun): masc gen sg (epic doric)

LSJ entries

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ἅλιος 1

ἅλιος (A), α, ον, also ος, ον S. Aj. 357, E. Heracl. 82 (lyr.): (ἅλς:—

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ἅλιος 2

ἅλιος (B), α, ον: (perh. cf. ἠλίθιος ):—mostly of things,

A. fruitless, idle, ἔπος, μῦθος, Il. 18.324, 5.715; πόνος 4.26; βέλος 5.18; ὅρκιον 4.158; in Od. only with ὁδός 2.273, 318; of a person, Il. 10.324: neut. as Adv., in vain, 13.505, cf. 4.179, S. OC 1469: reg. Adv. -ίως Id. Ph. 840.—Ep. word, used by S. in lyr.

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ἅλλομαι

ἅλλομαι, h.Cer. 175, etc.: impf. ἡλλόμην X. Cyr. 1.4.11, etc.: fut. ἁλοῦμαι (ὑπερ-) X. Eq. 8.4, Dor. ἁλεῦμ αι Theoc. 3.25, 5.144: aor. 1 ἡλάμην Batr. 225, E. Ion 1402, Ar. Ra. 243, no subj. or opt., part. ἁλάμενος [1st syll. long] Av. 1395, inf. ἅλασθαι Ael. Ep. 16, (καθ-) v.l. Luc. DMort. 14.5: aor. 2 ἡλόμην, rare in ind., v.l. S. HG 4.4.11, (ἐξ-) S. OT 1311, (ἐν-) v.l. A. Pers. 516, subj. ἅληται [ᾰ] Il. 21.536, opt. ἁλοίμην X. Mem. 1.3.9 (cf. εἰσ-), inf. ἁλέσθαι Opp. C. 1.83, etc., part. ἁλόμενος [ᾰ] A. Eu. 368 (lyr.), X. An. 4.2.17, etc.; to aor. 2 also belong Ep. 2 and 3sg. ἆλσο, ἆλτο, subj.ἅλεται Il. 11.192, part. ἄλμενος only in compds., but ἅλμενος Opp. H. 5.666: (sal-, cf. Lat. sal-io):—

2. c. acc., leap over, βόθρον Ael. NA 6.6; τάφρον Opp. C. 1.83.

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ἥλιος

ἥλιος, ὁ, Ep. ἠέλιος, as always in Hom. (exc. in the late passage Od. 8.271) and Hes., cf. Hp. Alim. 42: Dor. ἀέλιος [ᾱ] Pi. P. 4.144, Call. Cer. 92, Lav.Pall. 89, and lyr. in Trag., S. Ant. 809, E. Ph. 175, al., but ἅλιος [ᾱ], S. Tr. 96, E. Alc. 395 (ᾰέλιος S. Tr. 835): Cret. ἀβέλιος (i.e. ἀϝ-), Hsch.: Aeol. ἀέλιος Sapph. 79(= Oxy. 1787 Fr. 1.25), Supp. 25.7; ἄλιος Sapph. 69 (s.v.l.): Arc. ἀέλιος (or ἁ-) IG 5(2).4.12 (Tegea, iv B.C.):—

3. day, S. El. 424; a day, Pi. O. 13.37, Hp. Alim. 42, E. Hel. 652 (pl.), Ps.-Luc. Philopatr. 4,26, etc.; later, year, Herod. 10.1.

4. sunshine, sunʼs heat, ἐπὶ τοῖς ὄρεσιν Pl. Phd. 116e; ἥ. πολύς Luc. Nav. 35, cf.Herm. 25; πολὺν τὸν ἥ. ἐμφαίνειν, of a sunburnt person, Id. Ind. 3, cf. Rh.Pr. 9: pl., sunbeams, Thphr. Sign. 22, Ael. NA 16.17; hot sunny days, Th. 7.87.

5. metaph., sunshine, brightness, ψυχῆς Plu. 2.994e, cf. Artem. 2.36, etc.; of a person, Ἑλλάνων δόξης δεύτερον Ἀέλιον IG 14.1188; of Ptol. VI, UPZ 15.33; νέος Ἥ., of Nero and Caligula, SIG 814.34, 798.3.

II. as pr. n., Helios, the sun-god, Od. 8.271, etc.; νὴ τὸν Ἥ. Men. Sam. 108; ὑπὸ Δία Γῆν Ἥλιον, in manumission-formula, POxy. 48.6, 49.8 (i A.D.), IG 9(1).412 (Aetolia), IPE 2.54.10(iii A.D.); [Ἥλιος] δούλους ἐλευθέρους ποιεῖ Artem. 2.36; identified with Apollo, Carm.Pop. 12, E. Fr. 781.11; with Dionysus, D.Chr. 31.11, etc.

2. Ἡλίου ἀστήρ, of the planet Saturn, v.l. in Pl. Epin. 987c, cf. D.S. 2.30, Theo Sm. p.130H. (I.-E. sāwelios, cf. Cret. ἀβέλιος, Lith. sáulė, Lat. sōl.)