It is shown that the normal modal logic of two reflexive points jointed with a symmetric binary relation splits the lattice of normal extensions of the logic KTB. By this fact, it is easily seen that there exists the third largest logic in the class of all normal extensions of KTB.

We define a special family of Brouwerian logics determined by linearly ordered frames. Then we prove that all logics of this family have the finite model property and are Kripke complete.

Halldén complete modal logics are defined semantically. They have a nice characterization as they are determined by homogeneous Kripke frames.

We consider the family of logics from NExt which are determined by linear frames with reflexive and symmetric relation of accessibility. The condition of linearity in such frames was first defined in the paper [9]. We prove that the cardinality of the logics under consideration is uncountably infinite.

We prove that there is infinitely many tabular modal logics extending KB.Alt which have interpolation.