This essay starts with the basic concept and features of legal consciousness, adopts the methods of systematic analysis and that of the science of legal culture, taking legal consciousness as a sub-system under the social cultural system. In the first place, it makes a structural analysisto legal consciousness, revealing that legal consciousness is an organic whole with its own internal structure that can perform certain functions.
Then, the essay takes to discuss the functions and roles of legal consciousness as a system towards social economy, politics, and cultural system from the angles of economy, politics and culture respectively.
This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system.
In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.
The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables.
We consider 3-parametric polynomialsPμ*(x; q, t, s) which replace theAn-series interpolation Macdonald polynomialsPμ*(x; q, t) for theBCn-type root system.