  Two models with linear and nonlinear second class constraints are considered
and gauged by embedding in an extended phase space. These models are the free
non-relativistic particle on a hyperplane and hyper sphere in configuration
space. For the first model we construct its gauged corresponding by the
condition of converting second class system to first class one, directly. In
contrast the first class system related to the free particle on hyper sphere is
derived by the BFT embedding procedure, where its steps are infinite. We give a
practical formula for gauging linear and some of the nonlinear second class
systems, based on the simplified BFT method. As a result of the gauging two
models, we show that in the conversion of second class to the first class
constraints the minimum number of phase space degrees of freedom for both
systems is a pair of phase space coordinate. This pair for first system is a
coordinate and its momentum conjugate, but Poisson structure of embedded
non-relativistic particle on hyper sphere is a non-trivial one. We derive
infinite correction terms for the Hamiltonian of the nonlinear constraints. The
summation is done and constructs an interacting gauged Hamiltonian. We find an
open algebra for three first class objects of the embedded nonlinear system.
