  The nonlinear integral equations describing the spectra of the left and right
(continuous) quantum KdV equations on the cylinder are derived from integrable
lattice field theories, which turn out to allow the Bethe Ansatz equations of a
twisted ``spin -1/2'' chain. A very useful mapping to the more common nonlinear
integral equation of the twisted continuous spin $+1/2$ chain is found. The
diagonalization of the transfer matrix is performed. The vacua sector is
analysed in detail detecting the primary states of the minimal conformal models
and giving integral expressions for the eigenvalues of the transfer matrix.
Contact with the seminal papers \cite{BLZ, BLZ2} by Bazhanov, Lukyanov and
Zamolodchikov is realised. General expressions for the eigenvalues of the
infinite-dimensional abelian algebra of local integrals of motion are given and
explicitly calculated at the free fermion point.
