The paper is devoted to the study of the \(\nu\)-dimensional (\(\nu \geq 2)\) lattice models with spins taking values in any closed subset of \({\mathbb{R}}^ N\). The two body nearest neighbours interaction is assumed. The general method of investigation of the lattice stochastic Gibbs fields for the low temperature in the case of finite or countable number of the ground state minima is presented. The basic and essentially the only assumption is that the minima are sharp. The method is that of the contour distribution initiated by Peierls and developed by other authors. It is demonstrated that the case of continuous spin is not much more complicated than the discrete one so that the known procedure extends to this case. As a result all the cluster expansions are exponentially regular and give rise to the full cluster expansion of the transfer matrix. After presentation of the general method the particular cases are examined. a) Models with symmetries: i) Spins in a region or smooth manifold of \({\mathbb{R}}^ N\). Interaction described by a smooth function \(\Phi(s_ 1,s_ 2)\) with exactly two minima. The symmetry group is \(G={\mathbb{Z}}_ 2\) which exchanges the minima. The boundary conditions are fixed and constant. A particular representation for \(\Phi\) near minima is assumed. ii) Spins take any real values. \(\Phi(s_ 1,s_ 2)=m^{- 1}\psi(s_ 1-s_ 2)+\phi(s_ 1)+\phi(s_ 2)\), \(m>0\) and large, \(\phi\) is periodic with exactly one minimum inside the period, \(\psi(s_ 1-s_ 2)0\). Under additional assumptions on derivatives of \(\Phi\) it is shown that for sufficiently large \(\beta\) there is always \(\mu_ 0=\mu_ 0(\beta)\in U\) so that the system described by \(\Phi(s_ 1,s_ 2,\mu_ 0)\) has at least N different limiting Gibbs distributions. c) Spin in real segment. \(\Phi\) has one minimum and particular representation. Then for large \(\beta\) there is exactly one limiting Gibbs field which moreover depends analytically on \(\beta\).