What's so special about wave solutions of EM? Maxwell's equations allow for wave solutions via oscillations between electric and magnetic field content.
Couldn't we generate electric waves also if that solution didn't exists?
Imagine there was no magnetic field, only the electric field. If at one point in space we generated an electric field nicely varying in a sinus pattern of constant frequency, wouldn't this propagate with light speed, just like a plane EM-wave of that frequency, even though there was no magnetic field?
 A: No, it does not propagate. A low frequency electric field is a near field - it may depend on time but its distance-dependence is such that this field decays as $1/R^2$ at best. The propagating part of a low-frequency field ($\propto 1/R$) has a magnetic component too.
EDIT: The time-dependent field $E(t)$ has two terms: one is a near field and another is a propagating field $$E(t)=A_1/R^2+A_2\omega/R.$$ The propagating field strength is proportional to the frequency $\omega$ and is accompanied with a propagating magnetic field of the same strength. They can be small. A near magnetic field is smaller than a near electric field due to low frequency. So it is possible to have a variable electric field without variable magnetic field, but it does not propagate in reality. Light is a propagating EMF.
A: No. Why? Because the electromagnetic field is controlled by Maxwell's equations!
When you produce an oscillating electric field, it happens that the last of Maxwell's equation produces itself a magnetic field:
$$ \nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} +\;...$$
this means that whenever an electric field oscillates, a B field is generated in a certain way (and vice versa, as stated in another Maxwell's equation).
