  We study a time-inhomogeneous SDE in $\R^d$ driven by a cylindrical L\'evy
process with independent coordinates which may have different scaling
properties. Such a structure of the driving noise makes it strongly spatially
inhomogeneous and complicates the analysis of the model significantly. We prove
that the weak solution to the SDE is uniquely defined, is Markov, and has the
strong Feller property. The heat kernel of the process is presented as a
combination of an explicit `principal part' and a `residual part', subject to
certain $L^\infty(dx)\otimes L^1(dy)$ and $L^\infty(dx)\otimes
L^\infty(dy)$-estimates showing that this part is negligible in a short time,
in a sense. The main tool of the construction is the analytic parametrix
method, specially adapted to L\'evy-type generators with strong spatial
inhomogeneities.
