  A Hamiltonian formulation of generic many-body systems with balanced loss and
gain is presented. It is shown that a Hamiltonian formulation is possible only
if the balancing of loss and gain terms occur in a pairwise fashion. It is also
shown that with the choice of a suitable co-ordinate, the Hamiltonian can
always be reformulated in the background of a pseudo- Euclidean metric. If the
equations of motion of some of the well-known many-body systems like Calogero
models are generalized to include balanced loss and gain, it appears that the
same may not be amenable to a Hamiltonian formulation. A few exactly solvable
systems with balanced loss and gain, along with a set of integrals of motion is
constructed. The examples include a coupled chain of nonlinear oscillators and
a many-particle Calogero-type model with four-body inverse square plus two-body
pair-wise harmonic interactions. For the case of nonlinear oscillators, stable
solution exists even if the dissipation parameter has unbounded upper range.
Further, the range of the parameter for which the stable solutions are obtained
is independent of the total number of the oscillators. The set of coupled
nonlinear equations are solved exactly for the case when the values of all the
constants of motions except the Hamiltonian are equal to zero. Exact,
analytical classical solutions are presented for all the examples considered.
