Rukmini kalyanam was taken from 16th Century Telugu Poet Sri Bommera Potanas, Telugu translation of Sanskrit Srimadbagvatam, it contains around 108 slokas (Padyas) along with tika and tatparya (Meaning and Prose translation).
The actual procedure is:
You have to study all of the Rukmini Kalyanam slokas/padyas completely, (if possible with Prose translation and other simple puja like dhoop, deep and naivedyam to Goddess Sri Lakshmi) daily for 40 days. It may hardly take around one hour.
Rukmini devi pathra also known as Rukmini sandesam is in Shrimad Bhagawatham - dashama skandam - 52nd adyayam - shloka -37-43.
There is a belief that an unmarried girl reciting this would get married with a good god loving man. Don't worry about pronounciation. Shrimad Bhagawatham specifically sanctions anyone reciting the grantha with sincerety. It has been clearly mentioned in Shrimad Bhagawatham,
"Yasmin prathishlokam abhadhyavatyapi".
Meaning even if a soul recites Shrimad Bhagawatham by mispronouncing every shloka, God takes it kindly and bless his devotee.
Vyakyaanams have been written for these 7 shlokas for hundreds of pages.
Shloka composer is swayam jaganmatha.
She is sending the sandesha to Jagadeesha. Then you can imagine the content.
Rukmini says, hey! the most beautiful Lord, I have heard about your anantha kalyaana gunaas.
This is explained as the first step in bhakthi. Katha shravanam. To hear the leelas of Bhagawan. One gets Bhagawath ruchi on hearing the leelas of Bhagawan. That ultimately blesses him to reach his abode.
"kaale nrusimha naraloka manobhiramam"
Rukminidevi requests Shri Krishnaa to come as quickly as lord Nrusimhaa appeared for bhaktha prahlaada.
I am telling this is in the context of the belief, that unmarried girls reciting these shlokaas would get their marriage quite quickly.
Like Lord Krishnaa came quickly to take Rukminidevi with him, the divine couple bless the reciting unmarried girl for a quick vivaahaa.
Recite the seven sholkaas forth with. Don't bother about pronounciation.
With prayers to Radha Krishna yugalam to fulfill the wishes of their devotees.
Read Dasama Skandham in Bhagavatham- it is only reading Bhagavatham and rules attached to reading vrat for ramayana ,will be the same for bhagavatham.The other method is doing Rukmani Kalyanam in the Bhajanai padhthadhi by reading Ashtapathi. If no bhajans is attached, just reading Ashtapathi is okey.-consult a Bhajani Bhagavathar and take his guidance (Guru)
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