The Barut second-order equation: Lagrangian, dynamical invariants and interactions

The second-order equation in the (1/2, 0) circle plus (0,1 1/2) representation of the Lorentz group has been proposed by A. Barut in the 70s, ref. [1]. it permits to explain the mass splitting of leptons (e, mu, T). Recently, the interest has grown to this model (see, for instance, the papers by S. Kruglov [2] and J. P. Vigier et al. [3]). We continue the research deriving the equation from the first principles, finding dynamical invariants for this model, investigating the influence of potential interactions.