  We postulate the second-order derivative equation with four parameters for
spin-1/2 fermions possessing two mass states. For some choice of parameters
fermions propagate with the superluminal speed. Thus, the novel tachyonic
equation is suggested. The relativistic 20-component first-order wave equation
is formulated and projection operators extracting states with definite energy
and spin projections are obtained. The Lagrangian formulation of the
first-order equation is presented and the electric current and energy-momentum
tensor are found. The minimal and non-minimal electromagnetic interactions of
fermions are considered and Schr\"{o}dinger's form of the equation and the
quantum-mechanical Hamiltonian are obtained. The canonical quantization of the
field in the first-order formalism is performed and we find the vacuum
expectation of chronological pairing of operators.
