Invariant motions of particles for general three-dimensional subgroup of all spatial translation group

The equations of continuum mechanics are invariant under Galilei group extended by the dilatation. The group contains the Abelian subgroup of the space translations including the uniform motion of the origin (Galilei transformations). The Lie algebra of the group was studied and the optimal system of the subalgebras was constructed up to inner automorphisms. The Abelian subgroup of all space translations corresponds to 6-dimensional Abelian subalgebra, whose structure contains 13 dissimilar subalgebras. Among them there is the general 3-dimensional subalgebra containing all operators of the Galilei transformations. This subalgebra includes 5 arbitrary parameters which are the invariants of the inner automorphism group of the Lie algebra. For the general subalgebra we consider all invariant solutions with a linear field of the velocity for the ideal gas dynamics. The motions of particles are studied as a whole. The each particle moves on the straight line. The particles assemble on the linear manifolds of blow up at the specific times. There are some manifolds of blow up depending on the valuation of arbitrary parameters. The motions of isolated volumes from particles in the form of parallelepipeds projecting in the parallelogram on the manifold of blow up are considered. The movement of the sonic surfaces are studied for obtained solutions of gas dynamics equations depending on the state equation. The equations of the sound characteristics are introduced for the obtained invariant solutions. The example of the movement of a sonic conoid for a simplest solution is reduced.