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A mourner/Avel Davening for the Amud:
A mourner [during the first eleven months of mourning] does not lead the prayers on Rosh Chodesh, or on any day which has a Mussaf service. This restriction applies even to Maariv and Minchah of Rosh Chodesh.
If one has a Yartzite on Rosh Chodesh is he to Daven as Chazan?
Yes. This applies even if he is a mourner, within 12 months of mourning a parent.
 Darkei Moshe Y.D. 376/9 based on Binyamin Zev 161; Bach 384; Gr”a; Igros Kodesh 2/324; 8/236; 11/3; 24/149; Sefer Haminhagim p. 20 and 68 [English]; See P”M 108 in M”Z and 671 M”Z 8; Minhagei Chasam Sofer 1/14; Biur Halacha 132; Kuntres Ramach Osyos 225 [by Rav Shmuel Gronam] that so was custom of the Rebbe Maharash during year of Aveilus for Tzemach Tzedek
Other opinions: Some communities are accustomed that an Avel leads the prayer for all the Tefilos other than Musaf. [Gilyon Maharsha Y.D. in name of Kneses Hagedola 391; Opinion brought in brought in Igros Kodesh 24/149; Piskeiy Teshuvos 132 footnote 162] If this is the custom of one’s Shul or one’s community then he is to follow this custom and lead the prayers, even if his personal custom is not to do so. [Igros Kodesh ibid; See there that the person who Davened for the Amud was from Vilna and followed the Gr”a who ruled an Avel is not to Daven for the Amud on Rosh Chodesh. Nevertheless, he respected the community custom and did so a few times. Each time he began feeling really sick during Davening and hence asked the Rebbe if perhaps the cause for this is because he is breaking his custom of the Gr”a. The Rebbe replied that he is not to pay attention to this being that he followed the custom of his Shul and that he was obligated to do so according to some Poskim.]
 The reason: Some Poskim rule that an Avel does not lead the prayer on any day of joy in which Lamnatzeiach is omitted. [Noam Megadim p. 43] Some rule the reason is because on these days the congregation is in joy. [Maharil 22; Maharam Mintz 43] Others rule the reason is because on these days Gehinnom is not alit. [Binyamin Zev 161; Chabad custom]
 Poskim ibid; Rebbe ibid
 Igros Kodesh 14/430; Minchas Yitzchak 9/134