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I don't know much about Shanivar Vrat. The following advices are from web sites as given. Please consult an Astrologer or Temple Priest before starting Shanivar Vrat and do the Vrat according to their advice.
Please see this link to know about Saturday Fast
Saturday Fast : Lord Shani is worshipped on this day. Among the things offered to the Lord are Black Til (sesame oil), Black clothes, Oil, Urad (black gram whole). One should have meals only once in the evening after the puja is over. Food cooked should be without salt.Generally devotees worship Saturn idol made of iron. Black flowers, black sesame seeds (itls,), horse bean (urada) and black clothes are used for worship. Black articles and oil are given in donation. Black sesame and horse bean are taken as food. The Saturday fast keeps away all the hurdles and misfortunes. Lord Saturn is portrayed sitting in a glass chariot, yoked with eight black horses. People eat parched grams on this day to keep off the influence of Saturn. They massage their bodies with mustard oil and also give oil in donation. Lord Hanuman is also worshipped on Saturdays because disciples of Lord Hanuman are never influenced by the Shani sign. As the story goes, Ravana had held Lord Shani and Mahakal captive and it was Lord Hanuman who freed them from his captivity during the burning of Lanka. Feeling obliged, Lord Shani had promised Hanuman that he would not trouble those who worship Lord Hanuman.
Please see these links for Sri Hanuman Chalisa
Rituals for the Saturday Fast :
This story is related to Saturday fast. The devotees worship Saturn idol made of iron. Black flowers, black sesame seeds (til,). Horse bean (urada) and black clothes are used for worship. Black articles and oil are given in donation. Black sesame and horse bean are taken as food. Wives do not perform journey to their paternal hometown. The Saturday fast keeps away all the hurdles and misfortunes. Saturn sits in a mirror chariot, yoked with eight black horses. The recitation of Shani Stotra is also beneficial.
You can also see these links
Please see this link for Dasaratha Shani Stotram
Dasaratha Shani Stotras are considered an excellent remedy for Saturn related troubles. The following is from Skanda Purana.
Shani Shodasanama stotra
Kona shanaischaro mandah chhayaa hridayanandanah Maargandaja sudhaasouri neelavastraan janadyutih Abrahmanah kroorakroora karmaatangi grahanaayakah Krishnodharmaanujah shantah shushkodara varapradah
Apart from this reciting the following dashanama stotra of Shani by Dasharatha near a Pepal tree is also a highly recommended one.
Shani Dasanaama Stotra
Konastha pingalobabhruh Krishnoroudraantakoyamah Souri, shanaischaro mandah Pippaladishu sansthitah
Dasaratha Shani Stotram
Om. asya shree shanaishchara stotrasya dashratha rishi shanaish-charo-devata trishtup chandahah shanaish-chara preety-arthe jape viniyogah dasharatha uvacha
Om. Sri Shani Stotra, Dasharatha is the Rishi (seer of the mantra). Shanaishchara is god (related to this hymn). Trishtup is metre. Enjoined is repetition (of this hymn) for the appeasement of Shanaishchara (Saturn).
kono-antaka roudra-yama-tha babhruh krishnah shanih pingala manda sourih nityam smrito yo harate cha peedam tasmai namah shree ravi-nandanaya
Prostrations to Sri Ravinandana (Saturn), who when regularly remembered as (the ten names listed in the first and second line of this verse) steals away the suffering (misfortune of the devotee).
sura asurah kim purusha ragendra gandharva vidyadhara panna-gash cha peedyanti sarve vishama-stitena tasmai namah shree ravi-nandanaya
Prostrations to Sri Ravinandana, by whose unfavourable position, gods, demons, celestial beings, celestial musicians, celestial masters of various wisdom-teachings and even celestial snakes succumb to sufferings.
nara narendra pashavo mrigendra vanyas-cha ye keeta-patanga-bringah peedyanti sarve vishama-stitena tasmai namah shree ravi-nandanaya
Prostrations to Sri Ravinandana, whose unfavourable position brings agony to laymen, emperors, common animals, huge animals, forest creatures, insects, flies and humming bees (irrespective of their status).
vdesah-cha durgani vanani Yatra sena-nivesah pura-pattnani peedyanti sarve visham-stitena tasmai namah shree ravi-nandanaya
Prostrations to Sri Ravinandana, whose unfavourable position brings about suffering to countries, citadels, forests, war-camps, houses and cities.
tilairya-vair-masha gudanna danaih iohena neelambara danato va preenati mantrair-nijavasare cha tasmai namah shree ravi-nandanaya
Prostrations to Sri Ravinandana, who is appeased by giving in charity things like sesame, paddy, black-gram, molasses, rice pudding, iron, blue cloth etc., and by chanting mantras on his own day (Saturday).
prayaga-koole Yamuna tate cha sarasvati punya-jale guhayam yo yoginam dhyana-gatopi sookshmas tasmai namah shree ravi-nandanaya
Prostrations to Sri Ravinandana, who manifests in subtle form before the yogis who meditate on him at any holy bank of prayag (confluence) of two rivers; Yamuna and Saraswati, or in a cave.
anya-pradeshat swagriham pravishtas tadeeya-vare sa narah sukhe syat grihad gato yo na punah prayati tasmai namah sri ravi-nandanaya
Prostrations to Sri Ravinandana, who bestows happiness to a person when He enters His own house (constellation); (though) when He leaves the house, does not return for a long time.
srashta swayam-bhoor bhuvana trayasya trata hareesho harate pinakee ekas tridhah rig yajuh sama murtis tasmai namah sri ravi-nandanaya
Prostrations to Sri Ravinandana, who is the creator of three worlds, who is the self-manifested being, who is Hari the protector and Pinakin (Siva, bearer of the three pointed spear or pinaka), the destroyer, all in one, as well as the embodiment of the three-fold Vedas; Rig, Yajus and Saman.
konasthah pingala babruh krishno roudroantako yamah sourih shanaish-charo manday pippalandena sam stutah Thus, Pippalada praises Thee (by chanting your ten names as contained in lines 1-3).
etani dasha-namani pratar-uthaya yah pathet shanaishchara-krita peeda na kadachid bhavishyati Harih Om
He who chants these ten names, getting-up early in the morning, will never suffer from the pain caused by Saturn. Harih Om.