  Exact spectrum of the two-photon Rabi Hamiltonian is found, proceeding in
full analogy with the solution of standard (one-photon) Rabi Hamiltonian,
published by Braak in Phys. Rev. Lett. 107, 100401 (2011). The Hamiltonian is
rewritten as a set of two differential equations. Symmetries that get hidden
after further treatment are found. One can plainly see, how the Hilbert space
splits into four disjunct subspaces, categorized by four values of the symmetry
parameter $c=\pm1,\pm i$. There were only two values $\pm1$ for the standard
Rabi model. Four analytic functions are introduced by a recurrence scheme for
the coefficients of their series expansion. All their roots yield the complete
spectrum of the Hamiltonian. Eigenstates in Bargmann space are also at
disposal.
