This paper deals with the initial value problem of the type {Mathematical expression}
{Mathematical expression}where t is the time, L is a linear first order operator (matrix-type) in Quaternionic
Analysis and {Mathematical expression} is a regular function taking values in the Quaternionic Algebra.
The article proves necessary and sufficient conditions on the coefficients of operator L under which L is
associated to the Cauchy-Fueter operator of Quarternionic Analysis. This criterion makes it possible to
construct the operator L for which the initial problem (1), (2) is solvable for an arbitrary initial regular
function {Mathematical expression} and the solution is also regular for each t. ?? 2011 Springer Basel AG.
