Bivariate generating functions for involutions restricted by 3412

We study bivariate generating functions for the number of involutions in Sn subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern π or contain π exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.