  Quantum systems with constraints are often considered in modern theoretical
physcics. All realistic field models based on the idea of gauge symmetry are of
this type. A partial case of constraints being linear in coordinate and momenta
operators is very important. Namely, when one applies semiclassical methods to
an arbitrary constrained system, the constraints in "general position case"
become linear. In this paper, different mathematicals constructions for the
Hilbert space space for the constraint system are discussed. Properties of
Gaussian and quasi-Gaussian wave functions for these systems are investigated.
An analog of the notion of Maslov complex germ is suggested. Properties of
Hamiltonians being quadratic with respect to the coordinate and momenta
operators are discussed. The Maslov theorem (it says that there exists a
Gaussian eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian
system is stable) is generalized to the constrained systems. The case of
infinite number degrees of freedom (constrained Fock space) is also discussed.
