Bases of rearrangement-invariant spaces

We prove that if $E$ is a permutation-invariant space, then a boundedly complete basis exists in $E$, if and only if one of the following conditions holds: 1) $E$ is maximal and $E \ne L_1[0,1]$; 2) a certain (any) orthonormal system of functions from $L_\infty[0,1]$, possessing the properties of the Schauder basis for the space of continuous on $[0,1]$ functions with the norm $L_\infty$, represents a boundedly complete basis in $E$. As a corollary, we state the following assertion: any (certain) orthonormal system of functions from $L_\infty[0,1]$, possessing the properties of the Schauder basis for the space of continuous on $[0,1]$ functions with the norm $L_\infty$, represents a spanning basis in a separable permutation-invariant space $E$, if and only if the adjoint space $E^*$ is separable. We prove that in any separable permutation-invariant space $E$ the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable permutation-invariant space, if and only if at least one of the Boyd indices of this space is trivial.