Can a time variable B field create

[physics user1]

A static E Field and vice Versa?
Or a time variable field created also another time variable Field?

 

[cnh1995]

A static E Field and vice Versa?

Yes, provided the rate of change of one field is constant w.r.t. time.

 

[vanhees71]

The electromagnetic field is governed by the Maxwell equations. Written in terms of the usual three-vectors $\vec{E}$ and $\vec{B}$ (the electric and magnetic field components) there are four equations. Two of them are dynamical laws, describing the electromagnetic field as being caused by the charge and current distribution (i.e., on a fundamental level, by moving charged particles). The other two are constraint equations: One is Faraday's Law, and the other is Gauss's Law for the magnetic field:
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0.$$
The first one is sometimes misunderstood in the way that a time-varying (it doesn't matter whether the change is linear with time or not, by the way) causes an electromagnetic vortex field. Also Faraday's Law looks like this, it's physically a bit misleading to think in such a way. It turns out, that when you mathematically follow this idea that the equations for the solutions of the Maxwell equations become pretty complicated and non-local.

On the other hand, there are the Jefimenko equations, which are nothing else than the retarded solution of the Maxwell equations, which clearly show that the true sources in the sense of a local field theory are the charge and current distribution. For more details, see also

https://www.physicsforums.com/threads/induced-electric-fields.760783/#post-4792449