  We formulate the self-consistent separable random-phase-approximation (SRPA)
method and specify it for Skyrme forces with pairing for the case of axially
symmetric deformed nuclei. The factorization of the residual interaction allows
to avoid diagonalization of high-rank RPA matrices, which dramatically reduces
the computational expense. This advantage is crucial for the systems with a
huge configuration space, first of all for deformed nuclei. SRPA takes
self-consistently into account the contributions of both time-even and time-odd
Skyrme terms as well as of the Coulomb force and pairing. The method is
implemented to description of isovector E1 and isoscalar E2 giant resonances in
a representative set of deformed nuclei: $^{154}$Sm, $^{238}$U, and $^{254}$No.
Four different Skyrme parameterizations (SkT6, SkM*, SLy6, and SkI3) are
employed to explore dependence of the strength distributions on some basic
characteristics of the Skyrme functional and nuclear matter. In particular, we
discuss the role of isoscalar and isovector effective masses and their relation
to time-odd contributions. High sensitivity of the right flank of E1 resonance
to different Skyrme forces and the related artificial structure effects are
analyzed.
