Abstract: A process for obtaining atactic 1-butene polymer optionally containing at least one comonomer selected from ethylene, propylene or an alpha-olefin of formula CH2?CHRo, wherein Ro is a linear or branched C3-C20 alkyl group, comprising the step of polymerizing 1-butene and optionally ethylene, propylene or said alpha-olefin, in the presence of a catalyst system obtainable by contacting: a) at least one metallocene compound of formula (I) in its meso or meso-like form wherein M is an atom of a transition metal; p is an integer from 0 to 3; X, same or different, is a hydrogen atom, a halogen atom, or a hydrocarbon group; L is a divalent C1-C40 hydrocarbon radical optionally containing heteroatoms belonging to groups 13-17 of the Periodic Table of the Elements; R1 and R2, are C1-C40 hydrocarbon radicals; T, equal to or different from each other, is a moiety of formula (IIa) or (IIb): wherein R3 is a C1-C40 hydrocarbon radical; R4 and R6, are hydrogen atoms or C1-C40 hydrocarbon radicals; R5 is a C1