Suggest you check
http://www.telescope-optics.net/curvature.htm for the telescope types in question the aberrations are analytic to 3rd order at least from the Petzval sum.
Field curvature for refractors and 1 or 2 mirror systems is shown on the above page. The figure in the middle of the page quite nicely shows the situation for refractors vs Newtonians.
There is also a simple reason why this is the case - a simple doublet lens has a focal length that is almost constant as a point object (ie a star) as the object moves off axis the image of it must move in an arc - if it followed a flat plane the focal length must increase with the off-axis angle - which it doesn't - not for a doublet. The whole need for image flatteners arises from this.
For a Newtonian the simplest case is a spherical mirror - lets take a 6" f/8 for example which is near enough to parabolic to not matter. The spherical surface is symmetric about the centre of the sphere which means that if the primary is rotated about the centre to one side, the image MUST lie on a sphere of radius R/2 concentric with the centre of the primary spherical surface, ie concave towards an eyepiece and exactly as shown in the figure.
NOTE: Refractor = convex toward eyepiece, Newtonian = concave towards eyepiece. So in other words an eyepiece that suits one is not a great match for the other.
In a maksutov the field curvature is essentially determined by the two mirror radii, see
http://telescope-optics.net/MCT_off_axis.htm equation 133. Note the concluding remark: " fairly accurate estimate of the median image curvature in an MCT, in a typical system it can be expected to be numerically quite close to the secondary mirror radius of curvature."
It is also convex towards the eyepiece (same as for a refractor) because the Maksutov design is effectively spherically symmetric around the centre of curvature of the primary mirror - which you should know if you have a copy of Maksutovs original article or S&T Gleanings Bulletin C.
In an SCT field curvature is not affected by the corrector (zero power) and hence it is determined again as the petzval sum due to the mirrors and once again is similar to that of a refractor ie convex towards the eyepiece.
http://www.telescope-optics.net/SCT_...berrations.htm
See also
http://www.hnsky.org/LX200_optical_analysis.htm
Consequently a field flattener made for a refractor or SCT will not work in a Newtonian, and vice versa.