That Zabur (the Psalms) has been mentioned particularly in the last sentence of Al-Isra 17:55 ⧉: وَآتَيْنَا دَاوُودَ زَبُورًا ، (And We gave Dawud the Zabur) has been done, perhaps, because Zabur reports that the Holy Prophet ﷺ will be a ruler along with being a messenger and prophet. This is as it appears in the Qur'an: وَلَقَدْ كَتَبْنَا فِي الزَّبُورِ مِن بَعْدِ الذِّكْرِ أَنَّ الْأَرْضَ يَرِثُهَا عِبَادِيَ الصَّالِحُونَ (And We have already written in the Zabur, after the [ previous ] mention, that the land is inherited by My righteous servants - al-Anbiya', Al-Anbiya 21:105 ⧉).
At this stage, Imam al-Baghawi says in his Tafsir that Zabur is a book of Allah revealed to Sayyidna Dawud (علیہ السلام) . It has one hundred and fifty surahs or chapters and all surahs are restricted to praises of Allah and prayers to Him. The areas of lawful and unlawful, obligations and limits have not been taken up there.