Catalyzes the methyl esterification of L-isoaspartyl residues in peptides and proteins that result from spontaneous decomposition of normal L-aspartyl and L-asparaginyl residues. It plays a role in the repair and/or degradation of damaged proteins. [protein]-L-isoaspartate + S-adenosyl-L-methionine = [protein]-L-isoaspartate alpha-methyl ester + S-adenosyl-L-homocysteine Optimum temperature is 85 degrees Celsius. Highly thermostable, with no loss of activity after 60 min at 100 degrees Celsius. Enzyme activity is observed at temperatures as high as 93 degrees Celsius. Monomer. Belongs to the methyltransferase superfamily. L-isoaspartyl/D-aspartyl protein methyltransferase family.