This note is devoted to the construction and investigation of a fundamental solution (FS) of the Cauchy problem \[ (1)\quad \partial u(x,t)/\partial t+(Au)(x,t)+\sum^{m}_{k=1}(A_ ku)(x,t)=f(x,t),\quad x\in {\mathbb{R}}^ n,\quad t\in (0,T]; \]
\[ (2)\quad u(x,0)=\phi (x), \] where A and \(A_ 1,...,A_ m\) are pseudodifferential operators (\(\Psi\) DO) with symbols a(x,t,\(\xi)\) and \(a_ 1(x,t,\xi),...,a_ m(x,t,\xi)\) which are homogeneous in the variable \(\xi\) ; the principal symbol a(x,t,\(\xi)\) is assumed to be homogeneous of order \(\gamma\geq 1\) and elliptic: \[ (3)\quad Re a(x,t,\xi)\geq a_ 0>0, \] and the symbols \(a_ k(x,t,\xi)\) have orders of homogeneity \(\gamma_ k\), \(0<\gamma_ k<\gamma\).