https://books.google.com/books?id=3caCAAAAIAAJ&pg=PA225&lpg=PA225&dq="Tu+natale+solum"&source=bl&ots=p88nsRD-gm&sig=d-7HPCQTNSufRNSQAL4ZubXXc78&hl=en&sa=X&ei=iVTEVOebI8OngwSRs4LIDQ&ved=0CC4Q6AEwAg#v=onepage&q="Tu natale solum"&f=false

The Melody is in the Roman Antiphonal (Liber Antiphonarius, No. 820) 1949 ed.

I et II Vespers, Martinae celebri plaudite (4 verses)

Lauds, Tu natale solum (4 verses)

II Vespers in some places... Non ilam crucians (4 verses)

Now it gets confusing... according to Connelly,
The above are all part of one Hymn of 10 verses.
The arrangement of splitting the hymn as above was only done in SOME places. He gives V 1- 6 inclusive and last verse at Matins the last 4 verses at Lauds...
Further discussion can be found in this book, http://media.musicasacra.com/pdf/hymnsofbreviary.pdf

I note that the hymn is not in the L.U. but is in the online here, http://media.musicasacra.com/pdf/antiphonale-bw.pdf

Melody below, have checked the Global chant database and no alternative melody is suggested for this hymn.

FYI, I found change #44 from the 1960 Rubrics of the Roman Breviary and Missal, "Change in the Roman Breviary and Missal to conform with the new code of rubrics":

44. On the feast of St. Martina (January 30), the hymn Martinae celebri together with the second part Non illam crucians is said at Matins, while the hymn Tu natale solum is taken at Lauds and at Vespers.

Thanks to the analysis done by Kathy and Tomjaw, I'm assuming that the three hymns all have the same official melody, but if anyone has confirmation that would be appreciated.

The "Asclepiadic and Glyconic" 12.12.12.8 rhythm seems odd - not sure if I could even find a replacement.