Rule 1: 1. Three kinds of حَتَّى
As a preposition حَتَّى means 'until, as far as' and governs the genitive or a clause in the perfect: حَتَّى الْفَجْرِ، حَتَّى دَارَ مُحَرِّكُهَا. As a conjunction of purpose it means 'so that' and takes the subjunctive: حَتَّى لَا نُكَلِّفَهُ. As a particle of emphasis it means 'even' and has no effect on case: حَتَّى السَّيَّارَةُ لَا تُرِيدُ … 'even the car does not want …'.
دَفَعُوا السَّيَّارَةَ حَتَّى دَارَ مُحَرِّكُهَا.
dafaʿu s-sayyārata ḥattā dāra muḥarrikuhā
They pushed the car until its engine started.
'until', with a perfect.
خَرَجْنَا قَبْلَ الْفَجْرِ حَتَّى لَا نُكَلِّفَهُ شَيْئًا.
kharajnā qabla l-fajri ḥattā lā nukallifahu shayʾan
We left before dawn so as not to put him to any expense.
'so that', with a subjunctive.
حَتَّى السَّيَّارَةُ لَا تُرِيدُ أَنْ تَرْحَلَ!
ḥatta s-sayyāratu lā turīdu an tarḥala
Even the car doesn't want to leave!
'even': السَّيَّارَةُ stays nominative as subject.
بَقِينَا فِي الْقَرْيَةِ حَتَّى الْجُمُعَةِ.
baqīnā fi l-qaryati ḥatta l-jumuʿati
We stayed in the village until Friday.
'until', with the genitive.