Halakhah sa Isaias 59:20: Abudarham, Mishneh Torah, Prayer and the Priestly Blessing & Shulchan Arukh, Orach Chayim

And since this day is the day of judgment, all Israel is accustomed to arise early from the fifteenth of Elul on to say petitions and ask for mercy from the Blessed Name, until Yom Kippur on which judgment is sealed. And there are places where it is customary to rise early from Rosh Ḥodesh Elul, because on Rosh Ḥodesh Elul Moshe went up the mountain the third time to recieve the second tablets, and prayed on them for Israel, and his prayer was received, and he was given the tablets and descended on Yom Kippur, which was decreed a day of pardon and forgiveness for all generations. And the interpreters say that a source for this is Ani l-dodi v-dodi li (Song 6:3), which has the initial letters "Elul" and the final letters of four yods, adding up to forty, the number of days. And similarly, et levavkha v-et levav (Deuteronomy 30:2) has the initial letters "Elul." And similarly, u-va l-tzion goel ul-shavei (Isaiah 59:20) — the last two letters of goel an the first two letters of ul-shavei are "Elul." And Rabbeinu Hai wrote: "the custom is to say petitions on the Ten Days alone, though we have heard that in some local customs they rise from Rosh Ḥodesh Elul." And Rav Isaac ibn Ghayyat wrote: "as for us here, we rise from the beginning of Elul." And Rav Amram wrote: "On the ten days between Rosh haShana and Yom Kippur we arise early every day to the synagogues before morning light, and ask for mercy, and the prayer-leader begins with Ashrei and after that says Qaddish and begins with petitons and after finishing says "But we know not what to do" (II Chronicles 20:12) and finishes and says Qaddish.

After he has completed reciting the whole Amidah, he seats himself, bows his head, inclines slightly sideways; all the congregants doing the same. In this attitude, he offers up supplications. He then sits up and raises his head, the rest of the people doing the same; and, while seated, offers supplications aloud. The reader then rises alone and recites the Kaddish a second time, the Congregation making the responses as on the first occasion. He then recites, "And He, being merciful, forgiveth iniquity, etc.," then, the Psalm of David—(Psalm 145). The reader in the meanwhile stands; the congregation remains seated and recites with him. He then recites, "And a redeemer shall come unto Zion, etc. And as for me, this is my covenant, etc. But Thou art holy, etc. And one cried unto another, Holy, etc." He completes the Kedushah. The congregants respond, "Holy, holy, holy" three times. He then reads the Kedushah in the Aramaic version. Then he recites the text, "Then a wind lifted me up, etc." He reads this also again in the Aramaic version. Then he recites, "The Lord shall reign for ever and ever." This he also repeats in Aramaic, so that the people may understand.

“The order of the Afternoon Service¹⁸⁵Minḥah, מנחה, Afternoon Service; see footnote 40. on Yom Kippur” - Containing four paragraphs.
In the Afternoon Service one says “אשרי¹⁸⁶Ashrei, אשרי; see footnote 175.”, and “ובא לציון¹⁸⁷U-Va Le-Ẓiyyon, ובא לציון, is a prayer found near the beginning of the Minḥah Service on Shabbat as well as on festivals. It is of biblical origin coming from Isaiah 59:20-21. It usually follows the Ashrei (see footnote 175). It is also found in additional prayers often recited after the daily Shaḥarit, Morning Service. The prayer speaks of a Redeemer coming to Zion, Jerusalem, and helping Israel who eternally shares in God's covenant.
Hertz, op. cit., p. 571.”, but one does not say “ואני תפלתי¹⁸⁸Ve'ani Tefilati, ואני תפלתי, is a prayer recited on the Sabbath during the Minḥah Service. It closes the prayers immediately preceding the Torah reading done during the Afternoon Sabbath Service. It is only recited on the Sabbath. If a festival occurs on a weekday the prayer is omitted. On Yom Kippur the prayer is omitted even if Yom Kippur falls on the Sabbath because it is not in keeping with the spirit of the Day of Atonement. The prayer is from Psalms 69:14, "And as for me, may my prayer unto Thee, O Lord, be in an acceptable time: O God, in the abundance of Thy lovingkindness, answer me with thy sure salvation."
ibid., p. 572.” even if it (Yom Kippur) falls on the Sabbath.
Hagah : We are not used to saying “אשרי¹⁸⁹Ashrei, אשרי; see footnote 175.” and “ובא לציון¹⁹⁰U-Va Le-Ẓiyyon, ובא לציון; see footnote 187.” before the Afternoon Service, only before the Closing Service, Ne’ilah¹⁹¹Ne'ilah, נעילה, is a worship service which concludes the prayer service recited on Yom Kippur. The name Ne'ilah was derived from a ritual during the time of the Second Temple. Even though the service is now only recited at the conclusion of the Day of Atonement, it was originally recited on all public fast days including Yom Kippur.
The full name of the service is Ne'ilat She'arim which means the "Closing of the Gates". This referred to the daily closing of the Temple gates at night. On Yom Kippur this terminology is symbolically associated with the closing of the heavenly gates which are open until sunset to receive the atonement of a person who truly repents.
According to the Talmud, during Temple times Ne'ilah was recited daily one hour before sunset at which time the Temple Gates were closed as there was no evening sacrifice only an Afternoon Minḥah Sacrifice (see footnote 22). On Yom Kippur Ne'ilah, because of the length of the Temple Service and the special rituals involved, could not begin until close to sunset. After the destruction of the Temple and when Ne'ilah became only associated with Yom Kippur, it became the practice for the service to begin close to sunset, before twilight and to end at nightfall.
In the third century the Ne'ilah Service consisted of an Amidah of seven benedictions which were parallel to the other Amidot of the day (see footnote 43) and a confession of sins, the Al Ḥet (see footnote 45). The confession of sins though was later replaced by two prayers that are unique to the confession made during the Ne'ilah Service: Attah noten yad le-foshe'im, "Thou stretchest forth Thy hand (in forgiveness) to sinners," and Attah hivdalta enosh, "Thou has distinguished man (from the beast)." These prayers remind man that God eagerly forgives one who is genuinely penitent. There is a basic rabbinic idea that divine judgment is inscribed in the "Book of Life" on Rosh HaShanah, but it is not sealed until the end of the Day of Atonement. Therefore at the end of the Amidah in the Ne'ilah Service the root for the word inscribe, ktv, כתב, is replaced by the root for the word seal, ḥtm, חתם, when reference is made to the "Book of Life". To help set off the Ne'ilah Service and its uniqueness on Yom Kippur, the Ashrei (see footnote 175) and the U-Va Le-Ẓiyyon Go-el (see footnote 187) prayer which normally come at the beginning of the Minḥah Service on Sabbaths and festivals, come at this time before the Ne'ilah Service.
Beautiful liturgical poetry has become associated with the Ne'ilah Service especially prayers of forgiveness, seliḥot (see footnote 14). The central theme of the Service is to give one a final chance and effort to seek forgiveness before the heavenly gates symbolically close at sunset. The Service ends with a feeling of confidence.
The Ne'ilah Service contains piyyutim (see footnote 149), an Amidah and a reader's repetition of it, seliḥot, Ashamnu (see footnote 45), Avinu Malkenu (see footnote 15), and the Kaddish (see footnote 177). The entire ritual culminates with a responsive proclamation of the Shema once (see footnote 173, but only the one line "Hear, O Israel, the Lord our God, the Lord is One") followed by its response three times, Barukh shem kavod malkhuto le'olam va'ed, "Praised be His glorious Kingdom forever and ever", and Adonai Hu ha-Elohim", "The Lord He is God" (I Kings 18:39) seven times. Finally in the Ashkenazi rite a final single shofar blast (see footnote 221) marks the end of the Day of Atonement, the Sabbath of Sabbaths.
Herman Kieval, E. J., v. 12, pp. 943-44., and so wrote a few of our rabbis, (מרדכי והגהות מיימוני סוף ספר אהבה, וסוף הלכות יו״כ, כל בו ומנהגים).¹⁹²Mordekhai and Hagahot Maimuniyyot at the end of the book Ahavah, Love, (which is the name of the second book of Maimonides' Mishneh Torah, see footnote 59) and the end (of the section) of the Laws of Yom Kippur, and Kol Bo and Minhagim, ומנהגים מרדכי והגהות מיימוני סו ״ס״ אהבה וסוף הלכות יו״כ וכל בו.
For Mordekhai, מרדכי; see footnote 24.
For Hagahot Maimuniyyot, הגהות מיימוני; see footnote 27.
For Kol Bo, כל בו; see footnote 34.
For Minhagim, מנהגים; see footnote 13. We do not say “אין כאלהינו¹⁹³En Kh'Elohenu, אין כאלהינו, is one of the most popular chants connected with the Jewish liturgy. The Sephardim sing it everyday while the Ashkenazi rite has usually limited its singing to the end of the Musaf Service (see footnote 166) on Sabbaths and festivals shortly before the Aleinu Prayer (see footnote 182). During the geonic period (see footnote 19) it was also chanted during the Evening Services.
The prayer expresses God four ways: God, Lord, King, and Savior and that there is none like Him, who is like Him?, we will give thanks to Him, we will bless Him, and proclaim that He is God who our fathers worshipped. Each statement expresses the same thoughts repeated four times, once for each name of God. The first three letters of the first three expressions spells, אמן, Amen.
Hertz, op. cit., p. 544.”, (“there is none like our God”) on Yom Kippur, (מנהגים).¹⁹⁴Minhagim, מנהגים; see footnote 13.