Symmetry breaking for a system of two linear second-order ordinary differential equations (Q1577394)

For a system of two linear second-order ordinary differential equations the following canonical form is obtained: \[ x''=a(t)x+b(t)y, \qquad y''=c(t)x-a(t)y, \] and the Lie algebra of infinitesimal point symmetries depended on coefficients is calculated. The number of symmetries (dimension of symmetries Lie algebra) can be 5, 6, 7, 8 or 15. In particular it is shown that this system of equations is reducible to free particle equations \(x''=y''=0\) iff the number of symmetries is equal to 15 and in the case if the number of symmetries is 5 or 6 it cannot be reduced to a system with constant coefficients. For the oscillator-like equations \[ x''=-w(t)x,\quad y''=-w(t)y, \] that has 15 symmetries it is presented a transformation to the free particle equation.