  We consider deformations of $2\times2$ and $3\times3$ matrix linear ODEs with
rational coefficients with respect to singular points of Fuchsian type which
don't satisfy the well-known system of Schlesinger equations (or its natural
generalization). Some general statements concerning reducibility of such
deformations for $2\times2$ ODEs are proved. An explicit example of the general
non-Schlesinger deformation of $2\times2$-matrix ODE of the Fuchsian type with
4 singular points is constructed and application of such deformations to the
construction of special solutions of the corresponding Schlesinger systems is
discussed. Some examples of isomonodromy and non-isomonodromy deformations of
$3\times3$ matrix ODEs are considered. The latter arise as the compatibility
conditions with linear ODEs with non-singlevalued coefficients.
